Simply put, the loss of the American colonies was probably a benefit to the UK economy, and, if not a benefit, then a very mild loss. What's more, Britain was undergoing the Industrial Revolution at the same time, which was a massive increase in Britain's potential output. The American Revolution was a strain on the British government budget, as were the Napoleanic wars, but not an unbearable one.

R. P. Thomas and D. N. McCloskey have a useful summary of the research about the impact of North American (and other) colonies on Britain in their chapter Overseas Trade and Empire, note: pdf. They estimate that the benefit at most from the American empire was about £75,000 annually, out of about some British income of around £140million in 1770, while the costs to British taxpayers were about £400,000 annually. (Page 97). So from this point of view, American Independence was a net benefit to the British public.

Particularly as the American Revolution did not stop British trade with North America, exports to North America rose from a 6% share of total British exports in 1700-1701 to a 32% share of British exports in 1797-98 (note: these numbers are not comparable with the figures above as the foreign trade figures at the time were reported in fixed prices set in 1700 and thus are best interpreted as changes in volumes). While there wasn't a comparable rise in British imports from North America as a proportion, North American imports did keep their share of total British imports during that time.

Meanwhile, as I mentioned above, the UK was undergoing the industrial revolution. Daudin et al in 2008, note pdf summarised research into estimating the impacts of foreign trade on British GDP as "both approaches tend to produce small numbers, with the estimated contribution of empire or trade to growth remaining modest compared with the expansion of the domestic market (page 8). So that's the British economy side. How about the British government financing?

Great Britain did spend £12 million a year on the American Revolutionary War, which was small relative to the estimated total economy, but a large strain on British government finances. I've had difficulties finding figures on British government size in 1770, but in 1793 British government net revenue was estimated at around £10.6 million (table 8.4, British and French Finance During the Napoleonic Wars, note pdf, Bordo and White. In the same work, table 8.2 shows that in 1793, going into the Napoleonic wars, the outstanding British government debt was about £12.4 million. So the American Revolution was clearly a strain on government financing, but not an impossible one.

Bordo and White conclude that the British government had sufficient credibility as a borrower to finance the Napoleonic wars through borrowing, though the British government did raise taxes during the Napoleanic war and also run some inflation, rather than relying basically entirely on borrowing, as they did in previous wars (including the American Revolution).

So to summarise, American Independence in and of itself was probably a mild benefit to the British economy and thus to the British taxpayer. The British economy was taking off at this stage anyway with the Industrial Revolution. Fighting the war was a burden on the British government, but the debt had been mostly repaid by the start of the Napoleonic wars, helped by the growing British economy.

Sources: Chapter 4 in Floud and McCloskey, *The Economic History of Britain, 1700-Present *(1981), Vol. 1, pp. 87-102..

Daudin, O'Rourke, and de la Escosura, Trade and Empire, 1700-1870, note pdf, July 2008

Bordo, Michael D., and Eugene N. White. “A Tale of Two Currencies: British and French Finance During the Napoleonic Wars.” The Journal of Economic History, vol. 51, no. 2, 1991, pp. 303–316. JSTOR, JSTOR, www.jstor.org/stable/2122576.

Note: the Floud and McCloskey reference is fairly old but I'm not aware of any new research that overturns their results. If anyone is aware I'd be interested.

As a follow up I want to ask; Was there a very big financial loss to begin with? I always thought colonies were expensive, and trade would have continued during/after the war. If anything I would have thought losing the colony saved some money as troops were redeployed and colony maintenance decreased.