Being unable to move only prevents moving those 5 feet
The infestation cantrip states:
[...] The target must succeed on a Constitution saving throw, or it takes 1d6 poison damage and moves 5 feet in a random direction if it can move and its speed is at least 5 feet. Roll a d4 for the direction: 1, north; 2, south; 3, east; or 4, west. This movement doesn't provoke opportunity attacks, and if the direction rolled is blocked, the target doesn't move. [...]
If the rolled direction is blocked, the target doesn't move. This means the target doesn't move those 5 feet. If it somehow prevented them from moving at all, it would state as much, and if we do interpret this as preventing them from moving then no duration is listed and the creature could never move ever again.
Whose turn it is does not change
Nothing in the spell's description claims that moving (or failing to move) would end the creature's turn. You are simply forcing them to take some damage and move in a random direction, nothing else. The target's turn continues as normal after the spell is cast because no part of the spell states anything to the contrary.
The cantrip might use up 5 feet of the target's movement
This is a bit of a sidenote bug when this spell occurs on the target creature's turn, they would be forced to move 5 feet in a random direction. Notably, this movement is not using up their reaction or any other part of their action economy. Except, one could argue that this movement actually uses up their movement as it makes them move on their turn. One could also argue that it does not use up their movement, the same way it doesn't use their movement when it isn't their turn. This is a rather unclear section of the rules and the following question has answers arguing about a similar spell where you can see a few different trains of logic: