The issues to which you can legitimately raise objections have already been covered: Power and lighting should be split across both RCDs, so loosing one bank of circuits will ensure either power or light remains in any given area of the home. Cables (including incoming tails) should be suitably supported and restrained - there are many ways in which this can be done, including the simple cable clip. If there is sufficient space for finger access where tails come in the bottom, then this should be filled. Cables should be protected from metal edges as they come into the board, which can be remedied through the use of grommet strip.
The above are the ONLY issues that you could pull the original electrician on. As well as the bit of the circuit that has not been rewired. Be advised that you must give the original electrician the opportunity to fix those issues before going down the withholding payment or part payment route.
These are also the only things that you need rectified to make the installation totally compliant and safe for continued use. So as far as safety is concerned get the above fixed and you are fine to leave everything as it is.
You do not need a new board - It would just have been "better" if you had at least been given the option of an RCBO setup, But that ship has sailed as it were. There is zero recourse on the people who quoted for you when you simply asked for a fuse board upgrade or indeed if it was part of other works for not offering the RCBO option....It's just that most of us on here find it strange due to the fact that the cost difference these days is minimal and it is "best practice" but not really laid out in black and white in the regulations.
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