Do expats pay capital gains tax on share trading profits?

The short answer to this is no. Strictly speaking no-one pays capital gains tax on share trading profits as this would be subject to income tax and not CGT. Non UK residents are however exempt from CGT in any case. If a non resident was trading in shares they would need to be careful that they weren't classed as trading in the UK and as such subject to income tax. In practice though not only would it be relatively rare that share trading status would apply but also if you were resident in a country with which the UK had a double tax treaty, this would in any case restrict the right of the UK to tax profits.