In practical terms he can do anything. The unknown is whether the BBC (their lawyers) wouild [a] notice, and [b] think it infringes enough to [c] care and do something about it.
The word “TARDIS” is also trademark though, so has much the same risks.
My gut feel (and I want to stress again, I’m just some guy on the internet, and definitely not a lawyer) is that using the Tardis analogy of “bigger on the inside” and similar, is absolutely fine in discussions on the forums, most probably fine in documentation (even official doco), but using them overtly in banners or slogans is too risky