Hi there. We've had mouse problems before as well (live in a rural area). TBH if there is one there is usually at least a family of them. I gave up using humane mouse traps as it was taking too long to catch them all and sometimes they can die of fright trapped in them overnight anyway. If you set them free it has to be at least a mile from your home or they come straight back! So I found getting up at 5am every day and driving a mile just wasn't something I could keep up with. I went for the most humane killing solution I could find. Which are these - they work really well. They sedate the hamster almost instantly so they are asleep when the poison kicks in. It's fast - usually they are found either in the trap or 1 foot or two away from it.
I think it's more humane than a lot of other methods. Conventional mousetraps aren't instant and other poisons are awful. I do think you need to get rid of them quickly as yes they can pass on a disease to hamsters called Tyzers disease and it's very unpleasant. They are wild mice and do carry diseases.
Our local hardware store stocks these as well as Amazon so you might find some locally. You put them down along a skirting board where you think they have been.
https://www.amazon.co.uk/d/Home-Kitc...rds=Mouse+trap
What kind of cage is your hamster in? I always use 7mm bar spacing cages now just in case we get mice. Anything bigger than that, a mouse can get through. If your hammy is in a tank with a mesh grid top it should be fine. If your cage has 1cm bar spacing it might be an idea to move the cage to a room where you are sure there haven't been any mice in, and on top of a chest of drawers or something - away from curtains (curtains can be climbed!).
Also put one of the mouse traps in that room - but better make sure your hamster cage is escape-proof first and not have out of cage time until you've caught them. Except maybe in the dry bathtub.
They come in packs of two. I would get four. Put two in the room you know they've been in, one in a hallway or landing (under a radiator maybe) and one in the room you move your hamster to. If your hamster is an escape artist then don't put one in that room.