At 1:05:20 (see below), Thunderbird 3 is being positively peppered with space debris, some of which is bouncing off of the fuselage and engines. The engine fails to ignite shortly afterwards, well before the ship re-enters the atmosphere
As the ship pulls away from the station you see numerous small impacts and glancing blows from bits of Thunderbird 5. You can see a small plume of gas from near the affected engine (possibly from a retro burn but possibly from an impact) and they also manage to hit the station with one of the engines as they pull away. Any of these could conceivably have affected the engine's systems and caused it to fail.
On top of all that that, the ship was linked to the space station when the "Heat Exchanger" exploded. There's no corresponding images but the detonation appears to have blown away a large portion of the station's upper pylon. That debris (or the explosion itself) could also have been involved in the damage to the engine.