As Wot23 said - Spin a coin.
I think the general consensus among electrical institutions is that a rewire is recommended after about 30 years.
Beyond that - It's probably fine, but not everybody wants to put their name to it!
My late 1970's house (bought new by my parents, so I know the history) had the original fuse box replaced with an RCD unit about 5 years ago...
It was installed by a member of extended family who is a qualified electrician.
At no point was a complete re-wire advised or suggested. And there have been no issues whatsoever since installation - No "false trips" of the RCD.
It's been officially certified & was also inspected by whatever organisation inspects electricians work.
(they asked if they could put us down as one of the "X number of jobs per Y number of years" they have to have checked by an independent assessor as part of their qualification or trade body membership).
So everything "above board & legit".
However...
In the years since, a handful of sockets, light switches & ceiling light fittings have been opened for various DIY/Maintenance tasks and the wires, although still safe & sound - are showing signs of aging (the coloured sheaths around each wire starting to decay).
The house is structurally sound but is, in Estate Agent speak, "In need of modernisation" - I'm not planning on staying here forever, so while everything is working fine - I'll leave it as it is.
But if I were the next person buying this place - I'd be starting with a re-wire - Rather than get everywhere decorated, then have to rip it all out again when the time inevitably comes!
As for what would happen if you didn't (besides risk of fire or electric shock, depending on where the wire fails) - Only the circuit (wire) that's degraded would be affected & trip.
Circuits are typically (but not exclusively) done in groups, so:
"Upstairs lights"
"Downstairs lights"
"Upstairs sockets"
"Downstairs sockets"
"Cooker"
"Water Heater"
Whichever wire fails - that's the circuit you'll lose. But by that stage, if one wire has degraded to that extent - the others won't be far behind if they're all the same age.