I'm not looking for any solid answers, as I know it depends on the time spent exploring, but I'm just curious about a speculative answer or two.

Thanks!

Well, if it's a fairly good pet, and you can explore with it every day I'd say thirty days or so, because you end up levelling up once or twice a day that way, depending on number of battles won and so forth. You can probably fight the pet manually on arena morning and night and level it up fairly quickly that way too.

Training morning and night would probably take twenty five days, because each four-hour training session gives you one level. It's very expensive though.

Fighting on the arena or exploring and THEN training might level you up faster. Then again I've never been in too much of a hurry. My best pets I've just put on the arena via their profile pages and let them fight on auto. Free levelling.

I sometimes level pets manually in the Arena, and can say that it usually takes a bit under two weeks if you use a Sanctuary, and are lucky enough to have enough beatable opponents in the Arena every time (sometimes there's literally no other pets to fight). Absolute optimal speed using this method, if you always fight pets five levels higher than you (gives more experience), win every battle, and never let your energy max out without using it, is ten days. Of course you can make it even faster if you use energy drinks and/or train with the Ridinghall while waiting for your energy to regenerate, but I've never done either of those, since they're both rather costly. 

it depends how many times you take your pet on an exploration, and how many times you batle your pet.