I wondered how much energy could pass through some area 1000 miles away from the explosion, so I made some rough calculations.
Assumption: all energy of the nuclear explosion is radiation uniform in every direction.
The energy E would be spreading in a sphere with area 4*pi*R^2. Area A being a part of that sphere would capture E*A/(4*pi*R^2) of energy. So with:
energy E = 209 PJ (Tsar bomba for an extreme example)
area A = 1 m^2
radius of the sphere R = 1000 miles (1609 km)
That would give:
2.09 * 10^17 J * 1 m^2 / (4 * pi * (1.609 * 10^6 m) ^ 2) = 6.4 kJ
So at the distance of 1000 miles an 1 m^2 object would be exposed to 6.4 kJ of radiation. I don't know how destructive would that be. That energy would be equivalent to 6 seconds of sun irradiating a 1 m^2 area above the Earth's atmosphere, but of course sun's radiation is less dangerous (very little gamma/x-ray), and all of that radiation would be almost instant.
From what I'm reading about high-altitude nuclear testing, it can cause artificial radiation belts around the Earth (composed of high-energy electrons). I think that may be more dangerous to satellites at distances like 1000 miles (or at any distance) than the immediate gamma/x-ray radiation.