It would be a slow-moving but certain mass-extinction event
A single modern nuke is 30-80 times more powerful than Hiroshima
And then look at the socio-economic devastation of something like Covid
I'm not sure what's "a small number of nukes" in your book. Last I read about it, estimates suggest e.g. India and Pakistan dropping their entire arsenals on each other would indeed cause a nuclear winter.
In a nuclear attack, it's unlikely "we'll" be fine, but someone will be.
There never ever has been an above-ground test of a modern enhanced hydrogen bomb in the USA
The Bikini Atoll test in the 1950s using 75 year old technology killed people via radiation over 130 miles away
Modern ICBMs carry several much more powerful warheads than that
Even a few of them would make life on earth a permanent hell like peak covid pandemic but forever
Even the "smallest" nuclear war would be one-and-we're-done event
Eventually some tyrant on their deathbed will do it though
Holy shit.
Low LEO will mostly clear out in a few years. And for really important satellites that matter a lot no matter what the cost, they can still be used. You just have to accept a higher rate of attrition.
the micro-debris will make orbits impossible for the next century
and that's assuming they don't use EMP from orbit to take out ground communications too like starfish prime