Hello Ely,
I did my PGCE (11-18) three years ago, although I didn't last beyond the first term when I actually started teaching (workload too great, lack of basic supplies or support from school, etc.).
I don't think I would have had too much of a problem keeping up with my beekeeping tasks during most of the PGCE. Teaching placements don't usually kick off until the beginning of October, when you won't need to be doing much with your bees anyway, and the start is pretty gradual.
The only really difficult point would be from the middle of April to the middle of June (although you would get the May half term holiday). During this time the number of lessons you will need to plan is at its greatest, and you will also need to spend time making sure that you are putting together your Record of Development (a collection of evidence of all that you have done that shows your understanding and steps towards the Teaching Standards). At the same time you might also have to juggle writing some 'Professional Investigations' if your PGCE course includes masters credits.
Despite all of this, I think if you're only keeping one or two hives you shouldn't be too hard pressed to spare them an hour or two at the weekend. It will probably give you a bit of much needed light relief!
Keeping up with your beekeeping once you're teaching full-time will be the real challenge! I certainly don't think I could have kept up with that. As a new teacher, having to plan every lesson as you go, the work is non-stop; every night of the week, and most of the weekend. At least it was for me as a secondary science teacher having to plan Biology, Chemistry & Physics lessons and practicals for 7 different years.
Obviously everyone's experience will be different, but for me I would have struggled with keeping bees at least until I had a couple of years of full-time teaching (and a whole lot of lesson plans) under my belt.
Hope that helps a little