lets clarify what you mean..
are you saying that when you start the car the traction control light lights up on the dash almost immediately even without moving?
if so, what makes you think that the traction control stops working after 10 mins when the light has been on the entire time?
if the maf fails, (air flow sensor) the traction control light lights up and traction control does not work; while ABS is unaffected... sound familiar?
The engine will likely continue to run ok as it uses assumed figures and simply runs as slightly safer settings and likely a bit rich.
(im assuming its the same on the diesel as petrol)