As regards "podía", the point is that, in indirect speech, the present tense backshifts(*) to the imperfect, not to the preterite. In my dialect, "llevarle" is a case of "leísmo", so I'll use "llevarlo" (or "llevarla") in my explanation below. However, "llevarme" would make more sense to me in that context (the person wanted to know if the driver could take him or her somewhere):
- ¿Puedo llevarlo? (Can I take you?) => No tuve tiempo de preguntarle al conductor si podía llevarlo (I had no time to ask the driver if I could take him)
- ¿Puede llevarme? (Can you take me?) => No tuve tiempo de preguntarle al conductor si podía llevarme (I had no time to ask the driver if he could take me)
Notice that, just like "could" in English, "podía" in Spanish expresses past ability but not actual performance. The preterite "pude" or "pudo" would be equivalent to "was able to" or "managed to".
As regards the use of "tenía" or "tuve", both would be possible. "Tenía" refers to an ongoing state (I was not in a position to ask him) while "tuve" refers to an actual event (I had no time to ask him).
(*)Note: In indirect or reported speech, "backshifting" consists of retelling a person's statements or thoughts from a past perspective (e.g. Direct speech: Can you take me home? => Indirect speech: A asked B if he could take her home)