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The following explanations are from Dictionary of Phrase and Fable, giving the Derivation, Source, or Origin of Common Phrases, Allusions, and Words that Have a Tale to Tell (London: Cassell, Petter, and Galpin, [1870]), by the British lexicographer Ebenezer Cobham Brewer (1810-1897) [page 307, column 1, s.v. Fly]:

The fly on the coach-wheel. One who fancies himself of mighty importance, but who is in reality of none at all. The allusion is to the fable of a fly sitting on a chariot-wheel and saying, “See what a dust we make.”

This fable is as follows, in Fables, of Æsop And Other Eminent Mythologists: With Morals and Reflexions (London: Printed for R. Sare, B. Took, M. Gillyflower, A. & J. Churchil, G. Sawbridge, and H. Hindmarsh, 1669), by Roger L’Estrange (1616-1704) [The Fables of Abstemius, &c.: pages 244 & 245]:

Fab. CCLXX.
A Fly upon a Wheel.

What a Dust do I Raise! says the Fly, upon the Coach-Wheel? and what a Rate do I Drive at, says the same Fly again, upon the Horse’s Buttock?

The Moral.

This Fly in the Fable, is Every Trifling Arrogant Fop in Nature, by what Name or Title soever Dignify’d, or Distinguished.

Reflexion.

This may be Apply’d to well-nigh All sorts of Vain Persons and Humours: As Those that Assume to Themselves the Merit of Other Mens Services. Those that Talk, and Think, and Bussle, as if Nothing were done without them. All Meddlers, Boasters, and Impertinents, that Steal away the Reputation of Better Men for their Own Use. The World is Full, in fine, of These Pragmatical Flies, that Value themselves for being In at Every thing, and are found Effectually, at last to be just good for Nothing.
It is the Fortune, and it is the Humour of Weak and Trifling Men to Value themselves upon Idle and Trivial Matters; and many times, in Truth, upon Just Nothing at all: That is to say, upon a False Perswasion that they Do Things, which they do Not do, and Govern Affairs wherein they have No Manner of Interest. They Place a Reputation also upon Things that a Sober Man would be out of Countenance to Own, and Contend for the Credit of being the Authors of Fooleries. What a Dust do I Raise? says the silly Fly, And have we not Millions of Vain, Empty Pretenders in the World, that Talk at the same Rate, and with as Little Colour, either of Truth, or of Reason? ’Twas [I] carry’d such a Cause; such a Debate, such a Question. ’Twas [I] that Advis’d, Brought about, or Prevented This and That; when yet upon the Upshot, This same [I] was no more then the Fool, that fancy’d he play’d upon the Organ, when he only Drew the Bellows. Whence comes it now that Men Arrogate to Themselves thus, where they have Nothing to do, and Claim a Title, as Matter of Credit, to the Weakest Things in the World; but for want of Understanding the True Measures of Honour and Virtue: The Moral of This Vanity runs through All Degrees of Men, and All Functions. There’s Nothing so Great; There’s Nothing so Little, as not to Afford Subject for This Busie and Over-Weening Conceit to Work upon? No not from the Modelling of Common-wealths; The Winning of Battels; The Saving, or the Recovery of Kingdoms, to the very Flies Raising the Dust here in the Fable.

These are, in chronological order, the earliest occurrences of the phrase a fly on the wheel and variants that I have found:

1-: From The Authors Letter to an Anti-Episcopall Minister Concerning the Government of the Church, in Antiquity Triumphing over Novelty; Or, Presbytery Proved Popish, not Episcopacy (London: Printed for Philemon Stephens the Younger, 1661), by Hugh Edmonds [page 31]—the Latin phrase ergo hoc aliquid nihil est translates as therefore this something is nothing:

Neither can the force of the maxime be defeated by that small repulse of a great Rabbi in the schoole of Church Levellers, that one neighbour may compel another to give him satisfaction for wrongs offered, ergo hoc aliquid nihil est, which conceit is much like in feature to that ridiculous conceit of Æsops fly on the cart, who thought himself authour of the vapouring dust, which was raised by the violent motion of the wheels.

2-: From A Discourse of the Vanity of the Creature. By a Person of Honour (London: Printed by J. Macock, for Richard Royston, 1673) [page 20]:

’Tis not the Woman that reigns but the sin; ’tis not the Fly on the Coach-wheel that raiseth all this dust, but these trotting Lusts, which draw men to perdition.

3-: From Prodromus, or The Character of Mr Sherlock’s Book: Called A Discourse of the Knowledg [sic] of Jesus Christ. In which, The Evil Spirit, and Design, of that Book is Discovered, and several Errours therein, Confuted (London: Printed for the Author, 1674), by S. R. [page 22]:

You are but a spark, struck from the Forge of that Vulcan who made the first Thunderbolts that were to be used against the Non-Conformists. Don’t you think, like the fly upon the Coach-wheel, that you have raised all this dust.

4-: From An Answer to M^r. Cressy’s Epistle Apologetical to a Person of Honour Touching his Vindication of D^r. Stillingfleet (London: Printed by R. White for Hen. Mortlock, 1675), by the English theologian Edward Stillingfleet (1635-1699) [chapter 2, page 21]:

Come, come Mr. Cressy; let us not flatter our selves, it is not the Fly upon the Wheel that raises the Dust; we Writers of Controversies are no great Doers or Undoers of publick business.

5, 6 & 7-: From The Works of the Most Reverend Father in God, John Bramhall D.D. Late Lord Archbishop of Armagh, Primate and Metropolitane [sic] of all Ireland. Collected into One Volume. In Four Tomes (Dublin: Printed at His Majesties Printing House, 1676), by John Bramhall (1594-1663):

5-: [Tome 1, Discourse 3: A Just Vindication of the Church of England, page 178]—the noun privado designates an intimate private friend or confidant, especially the favourite of a ruler:

His next attempt is, to prove that the Protestants were the Authours of the Separation from Rome. And he names Three, Cranmer, Cromwell, and Barnes. […] It was the Universities that approved the Separation unanimously. It was the Synods that directed the Separation. It was the King that established the Separation. It was the Parliament that confirmed the Separation. How could two or three Privadoes, without Negromancy, have such an efficacious influence upon the Universities, and Synods, and Parliaments, and the King himself? Yet they might have an hand in it. No, nor so much as a little finger. As much as the Fly that sate upon the Cart-wheel, had in raising of the dust.

6 & 7-: [Tome 3, Discourse 2: Castigations of Mr. Hob’s Animadversions]:

6-: [page 817]:

He seemeth to ground much upon those Words which are added to the Last place, for the cause was from the Lord, conceiving some singular virtue to lie in them, and an ovation at least to be due unto himself, (I will not say, least the Bishop exclaim against me) applauding himself like the flie upon the Cart-wheel, See what a dust I raise, I will take the Liberty to tell him farther, That there is nothing of any cause of sin in the Text [&c.].

7-: [page 821]:

The World is too large a Sphere, and exceedeth the activity of poor little weak creatures, which are not able to Leave such an impression of might, as should move upwards to the convex superficies of Heaven, and downwards to the center of the earth, and round about to the extremities of the World. If this were true, the flie might say in earnest, See what a dust I raise.

8-: From L’Estrange a Papist, Proved by the Depositions upon Oath of Miles Prance, Mr. Lawr. Mowbray, […]. With several Animadversions upon the said Depositions, in Answer to Mr. L’Estrange’s late Pamphlets (London: Printed for Richard Baldwin, 1681), by Miles Prance (fl. 1678-1689) [page 22]:

To say truth, ’tis a very sorry Rhodomontado, the meer Fly upon the Coach-wheel.

9-: From Remarks from the Country; Upon the Two Letters Relating to the Convocation and Alterations in the Liturgy (London: Printed and are to be Sold by most Booksellers, 1689-90), by the Welsh theologian Henry Maurice (1648-1691) [page 7]:

Who could have imagined, that these little things should be of so great Consequence either for good or hurt? If these Trifles had but sense enough to understand the Charge layed against them, they could not but wonder with the Fly upon the Coach-wheel at the mighty Dust they made.

10-: From A Modest Attempt for Healing the Present Animosities in England. Occasion’d by a late Book, Entituled, A Modest Enquiry, &c. In a Dialogue between Testimony, a Zealous Dissenter; and Hot-head, a Chollerick Bigot; Trimmer, Moderator (London: Printed for R. Janeway, 1690) [page 17]:

Is this your Love to the Government, thus to bespatter its best and firmest Friends, who have contributed so much to our happiness, whereas Testimony! you were little more than the Fly on the Wheel, crying out, What a Dust do we make?

11-: From A Just and Sober Vindication of the Observations upon the Thirtieth of January and Twenty ninth of May (London: Richard Baldwin, 1694), by J. G. G. [page 31]:

For a Man so Zealous for the Cause only to wag his Finger, is but a small thing, except he thinks his little Finger stronger than another Man’s Loins; he thought with wagging his Finger to make all his Enemies afraid; […] is he not like the Fly upon the Coach-wheel crying out, See what a Dust I raise.

12-: From A Defence of the Short View of the Profaneness and Immorality of the English Stage, &c. Being a Reply to Mr. Congreve’s Amendments, &c. And to the Vindication of the Author of the Relapse (London: Printed for S. Keble, R. Sare, and H. Hindmarsh, 1699), by Jeremy Collier (1650-1726), anti-theatrical polemicist and bishop of the nonjuring Church of England [page 52]:

The fine Lady Cynthia out of her pious Education acquaints us, That though Marriage makes Man and Wife one Flesh, it leaves them still two Fools. But the little word Sᴛɪʟʟ is left out in the Quotation; which like the Fly on the Coach-Wheel, raises a mighty Dust. I grant I have by Chance omitted the word Sᴛɪʟʟ; and if he had done so too, the Sense had been perfectly the same, only better expressed. For Still is plainly useless, and comprehended in the Verb Leaves.