most of this isn't really directly about taxes but hopefully it's close enough

if not directly cited below numbers are in thousands of pounds sterling.

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The answer is it did have a real and negative impact but it was much much less than it could have been. One huge reason is that the long 18th century saw a steady expansion in the British state’s ability to raise revenue and get the relevant groups to buy into the government. Over the long 18th century (glorious revolution to waterloo) the government tax receipts expanded 18 fold while the GDP increased merely by a multiple of three. It’s not that it was no big deal to losing the war but in that type of increased extraction of tax revenue the question is more one of lower increases in future tax revenue than revenue contraction in real terms (talking over the medium run: I’m having trouble finding UK budgets in the short term. I know I’ve seen it referenced but I can’t find it).

It’s also worth pointing out that the revolution didn’t sever economic ties to great Britain, in fact the British rapidly returned to a strong economic position vis a vie the US. As P.J. MARSHALL puts it

The immediate recovery of markets in the United States and their subsequent rapid expansion suggested that the exports of British manufacturing industries did not need protection. The fact that Britain continued to act as the first destination for a considerable, if diminished, proportion of America's exports also implied that much colonial produce would come to Britain without compulsion.

Or, in the much more fun terms of Henry Clay the US were “'independent colonies of England', 'politically free' but 'commercially slaves'”. Read the whole speech here, interesting throughout To put some numbers on this british trade:

1770-1774 british (imports, exports) to 13 col/US: 1796, 3061

1784-1787: 845,2416

1788-1791: 1115, 3009

American Markets remained a major source of demand from Britain and a source of investment going forward. This was true often in the face of economic regulations on both sides: whether it was the Embargo act of 1807 in the US or the British navigation acts 1786 where the economic incentives of trade between the US and British colonial holdings existed attempts to keep each other out (though for different reasons) failed and were allowed to be not too strictly enforced before substantial modifications/revocations occurred.

There’s also the fortuitous timing of growing British control over India in the decade after the revolutionary war (featuring in part the defeated general Cornwallis). But that’s getting a bit off track and going way outside my comfort zone for the economics of that.

the loss of the colonies also mean the government was able to reduce the costs associated with maintaining the colonial frontier (that plus replayments from colonies for 7 years war debts are a huge part of the push for the stuff like the stamp act -> revolutionary war...but this is getting a bit off topic)

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other colonies

you can't talk about early british colonial history and ignores the Caribbean (which is deeply involved in how and why US grew the way it did). There's a whole different post on Caribbean stuff but I'll speak briefly on this.

West indies imports, exports

1770-1775: 3284, 1409

1788-1791: 3894, 2041.

with roughly 700 thousand pounds trade in import and exports between US and Carribean a year right before the revolution

by 1815 the brits are exporting 165k tons of sugar from the carribean.

while British east asia comes into her own in the 19th century East Asia still had a signficant economic impact.

In 1772/3 "East Indies" commanded over 20% of British imports and 1/7 of global british exports (2203k, 2096k)

that data is ultimately from: Phyllis Deane and W. A. Cole, British Economic Growth 1688-1959 (Cambridge, 1962), p. 87.