GPS satellites are in a Medium Earth Orbit, lower than geosynchronous. Each satellite orbits Earth twice per day. They're still really high up, at around 12,500 miles, whereas the highest ASAT test that I'm aware of was China's 2007 strike on Fengyun-1C, at just 537 miles of altitude.
The Starfish Prime nuclear test damaged quite a few satellites, but most of them were also much lower than GPS satellites - TRAAC at 590 miles, Transit 4B at 690 miles, and Ariel 1 at 247 to 747 miles in an elliptical orbit.
I think it's certainly plausible the satellites are still in orbit and transmitting, but they would lose accuracy over time. The atomic clocks aboard the GPS satellites drift by about 10 nanoseconds per day, which is equivalent to about 3 meters of distance on the ground. There's some randomness in that, so it won't be a steady loss of 3m/day of accuracy, but it would degrade over time. Galileo is roughly the same; GLONASS is an order of magnitude worse.
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The poster formerly known as The Dark
The Vespers War - Ninety years before the Twilight War, there was the Vespers War.
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