I'm a Brit and worked in the UK and US, both for the same employer on local contracts (i.e. not an assignment, though not sure that matters anyway) and was granted RSUs throughout this period. RSUs (at least for me) are taxed in two ways:
1) income tax (plus possibly other witholdings, like social security) when granted
2) capital gains when sold
My understanding is that if they vest while you're in the UK you'll pay income tax proportional to the amount of the vesting period that happened in the UK.
An example:
Jan 1st 2013: granted RSUs while working in US
Jan 1st 2014: move to UK
Jan 1st 2016: grant vests
In this case the grant spent 1 year vesting while in the US and 2 years in the UK. The US would tax 33% of the total value of the grant and the UK would tax the remaining 66%. Every grant to vest period is calculated like this. In my company this is all calculated and factored into the withholding as it vests. I'm not sure if this is universal.
If they're already vested then (I think) all that you will pay is capital gains tax when you sell. You should take advice here as I don't know whose CGT you would pay. In the US short term gains (< 1 year) are taxed as income, but long term (> 1 year) are at 15%. In the UK there is no distinction but there is an annual allowance of about £10k, then gains are taxed at 18% or 28% IIRC. I don't know if or how you can choose under whose tax jurisdiction these are taxed when sold.
I recommend asking your employer. My employer is very helpful in these matters and all this stuff is comprehensively explained on internal documentation. Also, if you don't have tax advice for your move it would be worth getting it. UK and US tax is complicated, to say the least. I'm not a tax professional and everything I say could be wrong
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