Any country capable of producing nuclear warheads will also be able to toss up enough BBs and other small objects into LEO to wipe out most of Starlink and anything else in LEO.

South Africa built nuclear weapons in the 1980s:

https://en.wikipedia.org/wiki/South_Africa_and_weapons_of_ma...

But it never had an orbital launch capability.

Pakistan doesn't have a domestic orbital launch capability but it does have nuclear weapons.

Surprisingly, the United Kingdom doesn't have a domestic orbital launch capability at present though it has had ballistic missiles and nuclear weapons for many decades.

At present, I would say that building a basic implosion-assembled atomic bomb is easier than building a rocket system that reach low Earth orbit. It's a lot easier to build a bomb now than it was in the 1940s. The main thing that prevents wider nuclear weapon proliferation is treaties and inspections, not inherent technical difficulties.

presumably the UK could figure out how to remove the top of a trident missile and replace it with a load of ball bearings

You do not need orbital capability to hit an orbital target. Just suborbital missile that reaches target's orbital altitude.