I’ve not seen that sort of carousel fitting to a door before.

Can’t you unscrew the carousel, add hinges to the carcase and then drill hinge holes to correspond to where you put the hinges so they line up on the doors?

Yes I think so. I think the carousel will have to be replaced too as it doesn't spin unless the doors are attached. I'm hoping that won't be too difficult though?!

Any DIYer should be able to do it. You can buy the hinges from Screwfix and Toolstation. Get someone to drill the hinge hole positions to correspond and job done. Then buy a new carousel if you want one. Depending on the specific one you choose I do find a lot of them are space wasting. I would just put a mid way shelf in there.

Instead of going to the effort of changing it why not get a kitchen fitter to reattach the doors to the carousel?

You need something like this. Mine is from Ikea.

If you want to change the carousel and have the doors hinged on the carcase you'll also have to add rails under the worktop and either a new bottom shelf or more rails, this is so the doors have something to stop against. At present the exisiting cabinet is bigger than the doors so the doors can fit inside and you don't want this if you want to hinge the doors. For the door hinging if you want to have the doors to bifold like the picture from Sarah L then you will need 2x 170 degree hinges and back plates to hinge the doors on the carcase, then the bi fold hinges and back plates to join the 2 doors together. I see that you also have long bar handles on the drawers so you don't want to hinge the door on that side as the door will get damaged by the drawer handles when its opened.