Revealing mistake: Throughout the film, whenever part of a wheeled vehicle lifts off the ground, the wheels stop moving instantly. In real life, the wheels would continue to turn at much the same rate, showing that these are unpowered models being pulled along.
Continuity mistake: When Thunderbird 1 prepares for take-off, it moves down a ramp and the launch site is seen at the end. It then cuts to a shot of it moving down the ramp, and then of it arriving at the site. The Thunderbird has suddenly rotated 90 degrees between shots.
Other mistake: Every time Thunderbird 3 is used during a mission, members of the Tracy family enter it by being lowered on the settee onto a mini rail system, which takes them to Thunderbird 3's hangar before a pole raises the settee into the ship. However, during the initial descent, there is usually an empty settee that passes behind the descending one carrying the impending Thunderbird 3 crew, to fill in the resulting hole in the floor. If the ascending one passes behind the descending one then there is no way that it could fill that hole as it would be too far back.
Suggested correction: As we see the replacement sofa correctly located, the fault is that we do not see the mechanism for doing this. For example, the poles might have been angled to achieve the effect or the start of the rail track might shift over a bit.
Continuity mistake: When Thunderbird 3 takes off from Tracy Island, look at the "3" at the bottom of the rocket when it goes up from the hangar. It is a completely different "3" to the one that was seen in the previous shot.