Fruit

You don't say how old the bushes are. If you move them, to give them a chance of establishing a decent root system you should probably pick off most if not all the flowers so their energy isn't spent on fruit. Make sure you prep the ground well and keep them watered while they establish again. If you buy plants, and there are lots of 2-year old potted plants available online, you probably won't get much fruit anyway in the first year, if any.

Strawberries should cope fine with being shifted, don't have any experience of raspberries yet.

To be honest I'd probably chance their survival and shift the lot, but quickly as everything's starting to grow.

As they are your plants you should be entitled to move them; however, if you know the new plot holder you could always ask them if they want them so as to avoid any problems. As said, it depends on how old and big they are and the effort in moving them successfully may be more than just getting new ones.

I would say transplant. I've just shifted a bed of raspberries and sorted out my strawberry bed. My blackcurrants are not yet in full leaf and no flowers to be seen, the same  applies with all of my fruit bushes other than the honeyberry bushes which are in flower.

I wouldn't pay for more fruit bushes when I could replant the ones I already have. You may risk a smaller yield this year but sometimes with raspberries and blackcurrants in my experience that's not a bad thing!   

I'm assuming that the new plot holder is not yet in possession of your old plot? If someone else is now working it I would think that you would now need their permission to take the fruit bushes. On my allotments anything that's left behind becomes the property of the new plot holder. Sheds, greenhouses etc included.