There's usually two transactions to consider when exercising stock options. Firstly, the exercise itself - the difference between the share price at exercise and the grant price is treated as employment income in both the UK and the US. Assuming you have been working in the UK throughout the period from grant to exercise, then the whole 'exercise gain' will be subject to UK tax, and your employer will have no choice but to withhold PAYE.
This will also be taxable in the US (as a US citizen) but because you've paid UK tax, you'll be able to claim it as a credit against the US liability, so the US tax should be zero (or close to it). If you've worked some time in the US this becomes a little more complicated.
The second transaction is the sale of the shares subsequent to exercise (if you sell them). The sale of the shares is a capital disposal for UK and US tax purposes, and if you exercise and sell on the same day, then this gain will be nil (or close to it). If you hold onto the shares, and the value increases, then there will be capital gains tax to pay on the increase between the share price at exercise (on which you have already paid income tax) and the share price at sale.
If the shares are non-UK shares, and you do not remit the proceeds of the sale to the UK, then this is where the remittance basis may come into play - you can shield the gain on sale from UK CGT.