Is there somewhere we can see how many days are left before a recently transferred pet can battle?

If you look at the pet's adoption date on his/her profile and subtract it by today's date then you can have a general idea of how many days are left until he/she can fight.

I have not adopted any battle pet so I can't think of any better method. Or perhaps you can try battling with him/her and maybe on the battle page it will say something about how many days are left.

Sorry I can't be of much more help :'(