There's a community allotment just down the road, which is well-kept and organised and has blooming flowers all year round.
Even if you can get all the allotment holders and the local parish council, you'll probably find that there just isn't enough on the allotments to keep your bees satisfied. Look further afield. Bees can travel quite a long way (~5 Km) so you shouldn't ignore hedgerows/ farm fields / etc.
One problem can be that, however hard you try, bees can and do sting people. This may cause your friendly allotment holders to become not so friendly. Another problem may be if/when they swarm. Can you deal with this in a timely fashion? If not, your allies may become increasingly hostile.
Finally, you hinted at some health problems that may impact on your ability to manage the colonies properly. I suggest that you think long and hard about this. Not only for your own safety, but, for those around you too. Bees need quite a bit of your time during the active season (March-October). If you can't provide this, you may be starting something that you can't maintain. Friends can be a great help, but, unless they are as committed as you are, you may find them less and less useful as time goes on.
I once had a couple of colonies near my local allotments. I thought, as you do, that they would do well off the fruit and veg being grown there. In fact, they were not very productive at all. One thing I hadn't taken into account was their need for shelter from any winds and the number of people working quite nearby. I thought this would be manageable, but, their presence seemed to attract inquisitive visitors.
It may work for you, but, my advice would be to find a site away from people if you can (possibly on a farm).