I think Marc refers to a few gotchas that I was explained recently after seeking professional advice. I had the same question a while ago as a PAYE french national non-domiciled in Ireland (I have no UK shares but U.S. shares instead with a U.S. brokerage firm bought before moving to Ireland)
1) Make sure the individual is really non-domiciled
2) Must be non Irish shares. Eg. the remittance basis doesn't apply to Irish companies traded in the UK,
3) If you fund a foreign credit card with the UK profits and regularly use it in Ireland this constitutes a remittance.
4) If you transfer the profits to your irish-domiciled irish tax resident wife's foreign account the remittance doesn't apply.
The remittance doesn't apply to the ETFs as they are largely domiciled in Ireland (not your case anyway) and other offshore investments like money markets, FX investments etc.
Other than this I was told that the remittance basis on shares applies (at least in my case) and need to declare and pay the tax on the dividends and CGT only if I remit to Ireland
Francois