What makes you think it might be ringworm?
At any rate DEFINITELY move him away from your birds.Not only are fleas, mites and ringworm contagious but just using the flea/tick spray around your birds is not good for them.
I've had hamsters before but never problem with mites or ringworm. My rabbit had ringworm before and my vet gave me some ointment to put on the affected areas. I was also told i could use over the counter ointment for athletes foot/ringworm etc. Like clotrimazole creme. Call your vet and see what he recommends
good luck with your hammie