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 19th August 1803, by a person signing himself Annibal, published in Cobbett’s Annual Register¹ (London) of 3rd September of the same year (Lloyd’s Patriotic Fund was created on 28th 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 16th 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 15th July 1815:
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.