Get a good, recommended and qualified electrician in (before making the appointment, ask if he is a member of a self-certification scheme - the correct answer is "yes" - and which one).
I think at the least you will need a new CU for the cooker to take the load off the old box; preferably transfer the electric shower as well, and, sadly, it's quite likely the whole house has an old electrical installation that's due for replacement.
If it's not convenient to do the whole house yet, consider having a large* split-load CU fitted, with capacity to add the other circuits to it as and when convenient. This could also be the best time to have a dedicated kitchen circuit installed, with loads of sockets and appliance outlets, to avoid disturbing the newly-fitted kitchen in future.
* the cost of a large CU, with spare, empty space in it, is very little more than the cost of a small one. You only need to buy some more fresh air and a little extra plastic.
I've just acquired a new cooker (not a range or anything like that) which states it must be wired to the terminal outlet with 6mm cable - I am guessing that this is an "at least" scenario and if there is anything bigger coming from the TO then it will be OK.
House was re-wired in Oct to very latest standards by the way.
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