I can recommend flypapers! I have them at the moment as we also have an outbreak of flies and I am not sure why (I suspect some rotting piece veg). Not from the hamster's cage but similar thing.
It was getting unpleasant. The flypapers got most of them in a couple of days. Some people may think it's a cruel way to kill flies but heck they're unhygienic and you need something the works quick. You can get a pack of 5 rolls on amazon. I put up 2 or 3 in different locations and they're virtually all gone now - the odd one still kicking about so I'm leaving them up in case any more hatch out!
For your hammy - I would do a full cage clean - they don't like it but hide some treats for him afterwards. If you can keep his nest and hoard in tact (check them carefully) it will help. And then don't put any fresh veg out for him until you're on top of the problem - I'm sure he'll be quite happy with something like half a walnut or a dried hamster treat etc instead of veg for a week or so. So there shouldn't be anything to attract flies to lay eggs in his cage. They will go for the fly papers instead (sticky honey things).
Get the rentokill ones - they work best. I am not like your housemates but we do get a box of fresh veg delivered fortnightly which I keep in the house as nowhere else to put it so suspect it was a rotting piece of veg that started it.
https://www.amazon.co.uk/Decco-Rento...981785&sr=8-12