Hi Under Par, I grow loads of bay trees in pots, mainly for resale down here in Cornwall. As you are in Bournemouth, it will be fine as regards hardiness!! I would suggest that the best time to repot it would be in the spring around the end of March and depending on how it looks, that would be the time to prune it or shape it as well. I would use any balanced compost, as long as its both well drained for winter and also that it holds the moisture in summer. One tip i can give you is to not fill the pot right to the brim, leave at least 2 -3 inches between the compost and pot rim so that in very dry weather, you can fill this with water to give it a good drink!! I use 'miracle grow' liquid feed on mine and give them a feed monthly but only up until end of july as i dont want them going into winter still making lots of soft growth that may be more easily dammaged by adverse weather! In subsequent years, I would prune them both in March and again the first week of August, that later trim helps to keep them very compact. Finally, the only other problem that you may encounter is that they tend to send up lots of suckers from the base, now if your growing it as a bush, this wont be a problem but if its on a clean stem, then these suckers can be a nuisance.
best wishes and good luck, Lannerman