Know what you're dealing with
First, the honest bit. One or two cockroaches that wandered in can be caught and released, and this quest shows you how. An established infestation (droppings like ground pepper, a musty almond smell, roaches visible in daylight) needs professional help, and no humane method changes that. UK infestations cluster around flats with shared walls, heating ducts, and bin stores.
Cockroaches are nocturnal. To find out what you have, go into the kitchen two hours after dark, flick the light on, and watch the floor edges. One scuttler is a visitor. Several are a colony, and you should skip to the last section.
The visitor scenario
Single roaches come in through gaps around pipes, under doors, in delivery boxes and secondhand appliances. Catching one is the same drill as a spider: glass over, card under, and steady nerves. They're fast but they can't fly out of a covered glass.
Wash your hands afterwards
Cockroaches can carry bacteria on their bodies. Handle them only via the glass, and wash up afterwards. Humane doesn't mean casual.
The jar trap for the ones you can't find
For a roach you keep glimpsing but can't catch: a jam jar trap. Take a clean jar, smear the inside of the top third with petroleum jelly (they can climb glass; the jelly stops them), and bait the bottom with a bit of banana or bread soaked in beer. Lean a lolly stick or strip of card against the outside as a ramp.
Set it against a wall in the kitchen overnight; roaches travel along edges. In the morning, take any catch well away from houses (a hundred metres or more, or they'll beat you home) and tip it out somewhere sheltered: a compost area, scrubby ground, anywhere that isn't someone else's kitchen wall.
Check the trap every morning
A humane trap only stays humane if it's emptied daily. An ignored trap is just a slow version of the thing you were avoiding.
Make the kitchen not worth the trip
Roaches need three things: food, water, and cracks. Remove all three and visitors stop returning. Food in sealed containers, no washing-up left overnight, bins with proper lids, pet bowls up after dark. Fix the dripping tap (there's a side quest for that) and dry the sink last thing; water matters more to them than food.
Then seal: caulk the gaps around pipes under the sink, fit a door brush strip, and close up cracks in skirting. In a flat, gaps around shared pipework are the main road in, and worth doing properly.
When to call it
If you're catching roaches nightly, seeing droppings, or smelling that musty note, it's a colony and it will out-breed any jar. Tell your landlord (in rented and shared buildings it's usually their responsibility) or book pest control. Getting there early is the humane option: a small treatment now beats a building-wide one later.