Hello. I am so sorry to hear this. It happens a lot and people aren’t informed properly about keeping pairs. A change of ownership and environment could make them fall out but so many other things can trigger it as well. Well done for separating then. I agree with the others it needs urgent vet treatment. But you really need an exotic vet as they know a lot more about hamsters. If you google exotic vet in your area there is bound to be one not too far away. They are no more expensive but your hamster will get a proper diagnosis and treatment.
Hopefully it will be a case of antibiotics and pain relief - it might need amputation so be prepared for that but they can do perfectly well without a foot. We have had 3 legged hamsters on here
. He will soon be thriving with love, attention and a bit of feeding up. If you’re looking for a new permanent cage the duna multy or hamster heaven are good options (providing all the plastic tubes and penthouse are taken out.