Pretty easily. Macaques live in much colder climates than Britain experiences, and they aren't half bad at coexisting with humans.
Indeed, before the last ice age there were macaques all over Europe. It's probably just bad luck that they didn't make it back before we arrived and started doing this civilization thing.
The only real obstacle for them would be that humans probably wouldn't let another feral animal loose in Britain. Especially if it could remotely be construed as dangerous. We can barely co-exhist with seagulls.
Barbary macaques, Japanese macaques, Tibetan macaques, Assam macaques and Arunachal macaques should have no trouble at all. Crab-eating macaques Stump-tailed macaques and Rhesus macaques would probably do fine too, especially in the south.