I'm thinking this is less a weed problem than a vegetable growing information problem. It's really hard to see what you've got there and impossible to id the weeds from the picture. Is that your seed bed? If so you need to check up the difference between things which you usually start off in a seed bed (cauli) and those which can be direct sown (carrots, radish, spring onions.) And always mark your rows so you can see where they are. If it's not your seed bed then its way too small for the mature plants.

It seems to me that you need to do a lot more research and reading if you want success in growing vegetables. It is not difficult but if you have never done it before you need to learn some basics. Look up each of the vegetables you have chosen to grow for how to start them off and look after them, prepare the ground thoroughly i.e. dig the ground and remove all weeds and then keep weeding constantly throughout the year. BTW cauliflower is one of the hardest things to get right.

You still have plenty of time this season. I have only planted out some lettuces (started indoors) and sown peas and broad beans so far. I am also in Somerset.

About the only big patch of green I can see, reminds me of purslane leaves. However, we really do need a better close-up to be sure of what you are describing.

Purslane is unlikely in the UK. The bushy thing looks like a clipped bay tree to me. But I can't see any other plants well enough to id them.

I applaud the effort you've made here, though I can't tell what I'm looking at, or what is growing either. Lots of people willing and able to help though, and even those who have a lot more info under their belt can have frustrations getting a veggie garden going. Doubtful I would have anything to add if Flora's 'on this,' I'm in Alabama, USA, but wanted to encourage you to keep up the good work, keep gathering info!