Killsats of
any kind will tend to increase the chance of reaching the Kessler threshold. The handful of conventional killsat tests already done, and the still very low number of unintentional sat-on-sat encounters, worries some people already.
And you couldn't surgically remove a foreign country's platform from space. Get rid of just the one spy-/monitor-sat (let's say by a hypothetical zero-debris EMP-pulse attack, to ignore the issue of an expanding/spreading cloud of schrapnel) just introduces a 'hole' in coverage that other current and imminently launchable replacements can fill (assuming the EMPing country isn't going to immediately assume it can use that gap to launch, which the EMPed country may also be able to assume). The single disabled satellite, assuming it isn't hardened enough to be at least reactivated, is now its own chunk of uncontrollable projectile, meaning that anything else out there has the full onus for adjusting to keep a safe path past/across its orbit. There may be secondary victims within range of the attempt to plug up a particular orbital keyhole, and
they might be as much of a problem, perhaps less hardened (civilian-type purpose, not intrinsically pre-hardened against attack, vulnerable whilst passing further away from the ambush position) and likely in similar-period orbits so might even be prime candidates to cross the same path half an orbit hence (or multiple: 1, 1.5, 2, ..., until the differences in resonance sends them 'apart' rather than potentially together), now pure space-junk.
Debris from the attacking kill-sat (and/or from the killed-sat, or ultimately any
subsequent collisions that might occur) are probably the thing to ultimately worry about (assuming the initial action, and what happens then on the ground, hasn't stopped anyone from caring too much about anything/anyone still sat above it all in Earth orbit), as it secondary, tertiary, quaternary(, etc) collisions then become increasingly inevitable. Or at least that's the theory. So far not realised (a handful of significant unintended occasions only, and several noted deliberate tests by the Soviet, US, Chinese and Indian (aero)space agencies, at least some of which set up using deliberately decaying orbits). Then there's the Starship Prime-like not-quite-spaceborne testing of potential anti-sat measures.
By today's standards of space-cooperation, it seems that Russia is the most likely to press such circumstances (with conventional attacks against their own 'test' targets) with perhaps eight active tests over the last decade, plus active manouvering of 'shadowing' craft (to relatively close proximity of actual US hardware). Though maybe the US/etc has just done enough to know that their hardware is capable, and the other nations with a space-launch capability have satisfied themselves about their ability to 'thread the needle' and are keeping their full hand close to their chest.
Nuclear material in space isn't that surprising, with numerous RTGs scraping by, and I'd be very surprised if there haven't been more
actual weaponised deployments than anyone suspects, perhaps midway between sensible RTG amounts of material and 'full-blown' bomb-sized. But, as with ground warfare, first use of actual nuclear weapons will be a "first-strike of last-resort", with kinetic attacks of various kinds being more easy to fulfil and 'justify' without
necessarily tripping things over into WW3 (though it wouldn't really help stop that, either).