I think the others have told you what you need to know, Arrods. Naturally, it all depends on the total load.
From your second post, it seems as though one of the ovens can use the existing cooker circuit, and the other can be plugged into a ring circuit which, in the case of rewireable fuses, will be a 30A circuit.
Information plates (usually somewhere on the back of the oven) will give the load. Your electrician will soon be able to decide which is the best way.
As far as the consumer unit is concerned, there is no requirement for an upgrade unless, for some reason, the present arrangement is unsafe. It all costs money, but a modern CU is reccomended. It would be a good investment. Bear in mind though, that it would make sense for the entire system to be tested before upgrading.
Older rewireable systems can be quite tolerant of minor faults. A new CU with overall RCD protection (earth leakage trip) may well show up such faults. I've made this upgrade on several occasions without any problems, but you never can tell. All systems should be checked every 10 years. It would be in your interest to have this carried out.
In short - modern CUs have MCBs (miniature circuit breakers) in place of fuses. Much more convenient to reset. Your fuses may well be safely protecting the system from fire etc, but you have no RCD. Roughly speaking, MCBs protect the cables etc, while RCDs protect you.
The other "biggie" is your existing Earthing arrangement. Essential to have this checked too.