I've been pondering over this post for a while and trying to work out how best to answer. I could just post the order as I would do it direct from your list:
2. Beans & peas
4. Aubergine, capsicums & tomatoes,
3. Brocolli, Chinese brocolli, Chinese cabbage & Pak Choi
1. Beetroot, carrots, celery, leek, onion & spring onions
but it's leaving me with an itchy finger to add more
I think what is puzzling me is that there are a lot of different types of veg - so either you are only growing one or two of each, or the beds are a lot bigger than I imagine (in which case, maybe you should "sub-divide" into more?)
The pea & bean bed is fine - if your soil pH is a bit on the acidic side, add a little lime as this will help nitrogen fixing bacteria to work properly.
The aubergine might be better in a greenhouse (so would the capsicums & tomatoes for that matter, but only because we've had a couple of cool summers that have given disappointing results) Definitely dig in some extra compost or manure into this bed for good results.
Celery needs more food than carrots, as do beetroot (in my opinion), so they might be better in the aubergine/tomatoe bed - but my concern is their growth can be quite sprawly or bushy which might affect the celery & beetroot.
Adding lime prior to planting the brassicas raises the pH (helps to prevent clubroot), and if you have heavy clay, helps to make it more workable by the time the carrots are growing the year after.
The alliums are a group in themselves, but many people do like to companion plant them with carrots in an effort to confuse carrot flies so perhaps leave them if you want.
Hopefully if I bump this up, someone else might see the post & add to it (just to confuse you more
)