I'd be perfectly fine with a female Time Lord show...but that said, I don't want the Doctor to be a woman. Where, in the Doctor Who mythos, is it established that Time Lords can swap genders when they regenerate, anyway? (Besides the "Curse of the Fatal Death" comedy episode, which is hardly canonical.) [Disclaimer: I haven't seen the most recent season of the show - I'm waiting for Christmas to get it on DVD - so hopefully it isn't established in one of those episodes, as I'd hate for that concept to become the norm.]
To me, the Doctor needs to be male - and furthermore, British. (I'd be as displeased with an American Doctor as I would with a female one.) If nothing else, if Gallifreyan gender-swapping were as prevalent as some would like to think, why has the Doctor remained male throughout all of his eleven incarnations thus far? Likewise with the Master, likewise with Romana (who was female in all of her incarnations), likewise with Borusa and every Time Lord shown thus far?
As for concurrent TV shows, I believe over in the Star Trek universe we had "Deep Space Nine" starting up while "Star Trek: the Next Generation" was still in production, and likewise "Voyager" started up while "Deep Space Nine" was still running. So it's been done before, and I'd be perfectly fine with a separate female Time Lord TV show running at the same time as "Doctor Who." That way, we crusty old traditionalists get to keep our male Doctor and you can still have a female Time Lord running around giving the Doctor a run for his money on her own show.
Johnathan