The phrase backward in coming forward means reluctant, shy to do something; here, backward means unwilling, and to come forward means to volunteer to do something.

The earliest instance that I have found, in the form backward to come forward, is from a letter dated 19^th August 1803, by a person signing himself Annibal, published in Cobbett’s Annual Register¹ (London) of 3^rd September of the same year (Lloyd’s Patriotic Fund was created on 28^th July 1803, during the Napoleonic Wars, to give grants to those wounded in service to the Crown and to set up annuities to the dependents of those killed in action):

In the Morning Chronicle of the 16^th inst. there appeared a paragraph respecting the subscriptions to the fund at Lloyd’s, which paragraph I must consider, as a libel on the nobility and gentry of the country; and, I shall, accordingly, trouble you with a few observations upon it. […] The nobility and gentry who have not set down their names to this subscription […] at least forbear to bring into public light and notoriety, the mortifying fact of the inferiority, in point of opulence, of the landed gentry to the merchants. […] Gentlemen of eminent families may well and properly be grieved, if they were to see published in every ale-house of the land, their inability to equal, in pecuniary sacrifices, those who may be of very inferior origin. […] Is it matter of surprise, that men proud of their birth, of their hereditary honours, of their ancient families, should be backward to come forward under these auspices?

¹ founded by the pamphleteer and journalist William Cobbett (1763-1835)

Curiously, the context of second-earliest instance that I have found, in the form backward at coming forward, is also a subscription opened in order to provide aid to soldiers wounded during the Napoleonic Wars; it is from The Lancaster Gazette (Lancaster, Lancashire) of 15^th July 1815:

LANCASTER,
JULY 15, 1815.

We will readily give place to “A Fiddler’s” promised observations on the propriety of opening a subscription in this town, in aid of the fund already raised in London, for the relief and benefit of the sufferers in the battle of Waterloo; but we trust that, before our next publication, measures will be adopted to supersede the necessity of their insertion. “The Lads of Lune²” have never yet been backward at coming forward on similar occasions, and we perfectly agree with our correspondent in thinking, that this is not a proper time to begin a new line of conduct.

² Lancaster is on the estuary of the River Lune.