It is (just about) possible to find bikes that are British-made in part, but is it a pointless exercise or a good use of your money? If you owned a bike before the 1980s in the UK, the chances are it was British-made. If you own one today, it most likely isn’t. Read article
No much chance of an Irish one either, but check out Men’s Sheds; they’re doing work on second-hand bicycles …