Depending on how dry the wood is, I would wait a week or so, depending on the temps. I am staining a redwood patio table right now and it is sucking up the stain like crazy. If the oil is not absorbed all the way the stain will not take well at all.
Be careful too, rodents will chew on wood frames and you need to be careful about toxicity.
I wouldn’t put the mice in the cage for several weeks after finishing the cage.

I’m using a nontoxic varnish that’s also suitable for children’s toys, but I’m still going to wait for a few days before I put them inside, because of the fumes and the time it takes to dry of course- thank you for your answer, will wait for a week.

@scuniper Oh wow..non-toxic stain…great! :-)
I had a parrot once that chewed on a freshly stained dowel on my staircase, he was fine, but I was so freaked out, called the vet. He escaped his cage and decided that the dowels on my staircase were really fun to snap in half. lol

It was recommended in all books and articles I read to up my knowledge on mice, as they do like to nibble wood, much like your parrot ;) And god, that must look rather interesting, dowels snapped in half by a parrot’s beak ;)

Different products require different times. I would just read the can.

I ususally oil or varnish, not both. Will the varnish stick on top of oil? Will you sand it first? Let me know how this works out, I am intrigued.

I’ve used a wax rub, sand, shellac, steel wool ( two coats wooling between ) and a spar varnish ( two coats wooling between too ) on wood with a great look on grain.

I gave the oil 2 days, it was dry and hard by that time. Then I varnished right away, no sanding. Tomorrow I’ll add a second layer of varnish, but it’s going well so far.

If it were me I wouldn’t oil, (stain?) and varnish a cage that a chewing animal is going in. Leave it plain wood, because that guy is going to spoil all your work. Actually wood is probably not the best material unless you can line it with a removable liner to protect it. Was this a commercially produced item or something home -made?

It’s commercially produced, but with mice wood seems to be the best material available, at least that seems to be a wide-spread opinion under mouse holders. The cage is oiled and varnished because it would get soaked with the mice’s urine otherwise, which smells rather bad. And only the floors are oiled and varnished- they won’t be able to chew on them ;)