Did you find any issues in having a player character Time Lord? Issues I've heard of in other games have ranged from the Time lord being 'more powerful' than other characters, it being a bad idea to have a time machine in the hands of a PC, and the issue of how do you handle the Time lord's knowledge without him having to repeat what you just told him-the Galaxy Quest "I have one job on this spaceship.." problem. Faced with these issues, the players i deal with actually prefer an NPC Time Lord. I'm curious what your thoughts were.
A Time Lord PC provides a level of power and responsibility I wouldn't give to just anyone. I was happy to go for it in this case, as the group as a whole nudged Kai's player towards the Time Lord role because (a) he's a card-carrying Who nut who (b) he has an uncanny ability to roll badly on crucial dice rolls so having a character who could regenerate from mortal injuries seemed smart, and (c) he's a GM himself and both able to come up with his own bizarre technobabble and from previous form demonstrably unlikely to go on a mad power trip.
In a few cases I prompted some particular info or technobabble in advance, in some cases I explained what something was and he said "I say that!" and in some cases he came up with his own improvised space history and pseudoscientific gobbledigook.
(And in terms of raw power at the table, everyone could do something better than Kai, Gabriel edged him out by a point or two in Ingenuity + Technology rolls, and the telepathic and telekinetic Nimue could wipe the floor with him.)
The TARDIS, by comparison, was no trouble at all. It was occasionally used as an emergency teleport (powered by Story Points, so it's a specific variation of using Story Points to dodge trouble) but it being a time machine really only featured in the final episode. Otherwise it played the traditional role of dropping the characters into the story at the start. The group as a whole had some say in where the TARDIS dropped the characters, rather than Kai's player in particular.
The only times when it might have had a greater impact were in adventures expressly about time travel. In Ghost Ship it was almost immediately written out, while in The Girl That Time Forgot it was used on Effie's behalf.