a few hundred, maybe, and back in the day they had thousands.
hit the 10 largest cities and it's basically over. big cities are also primary transport hubs of food and fuel, and with those gone everything else collapses. most people aren't farmers, and even if they were, no one is using pulled plows in the First World these days, so without gas and farming everyone starves. most of your best educated, most likely to govern smartly, are also in those 10 big cities; everything turns into Riddley Walker pretty quick.
the US or Europe or Russia or China are a big larger, but that just means you need 20-40 instead of 10. 100 nukes is enough for basically all of the West, or Russia, or China, etc. 1000 if you want to be sure, and have some redundancy / second-strike capability.
I haven't inquired about the UK, but that is not even close to true for the US.
For one thing, at any given time, there's enough food stored on US farms to feed half the US population for about 3 years, which is probably enough time to restart mechanized agriculture or failing that re-open enough port facilities to import enough food from our friends to keep most survivors alive.
(This food stored on farms is mostly intended to be fed to farm animals, but it is food humans can live on even if they probably cannot thrive on it.)
A nuclear attack leaves most internal-combustion vehicles intact. The US produces all the oil it needs, and the attack necessarily leaves most of the wells intact because (like the vehicles) the wells are too spread out for an attack with even 3000 warheads to get even half of the wells.
The vast majority of comments on nuclear war on the internet are wrong, and it offends me that people are being so careless about spreading falsehoods. (Spreading these falsehoods does not make us safer.)