The interior isn't in a "different dimension".
It's just bigger on the inside, thanks to Time Lord science -- to wit, dimensional engineering.
As the Fourteenth Doctor points out when someone states you can't "engineer dimensions", "maybe you can't."
It should be noted that throughout the entirety of time and space, the Time Lords seem to be the only ...
The Doctor explains it himself, during The Time of The Doctor (emphasis mine):
DOCTOR: I knew. I always knew it wasn't over.
CLARA: What is it?
DOCTOR: A split in the skin of reality. A tiny sliver of the 26th of June, 2010. The day the universe blew up.
CLARA: Missed that.
DOCTOR: I rebooted it, put it all back together.
'What dimension is the interior of the TARDIS in'?
The short answer is: the same dimension that the exterior is in.
Remember that the true meaning of 'dimensions' (which is what the 'D' in TARDIS stands for) is quite different to the common sci-fi usage of the word 'dimensions' to refer to other realms or parallel universes. There is, obviously, a ...
A "dimension" is an orthagonal direction, not an "alternate universe". The 3 special dimensions that you are most used to are Longitude (x-axis, East/West), Latitude (y-axis, North/South) and Altitude (z-axis, Up/Down). The Tardis extends into an additional dimension - spissitude (w-axis, Kata/Ana) - of which we are not typically aware.
The Tardis is to ...
The obvious, out-of-universe answer to this question is that each regeneration shown in the early TV show was an example of the visual special-effects of the day. The show has spanned so many decades it is no wonder that the SFX have improved. The newer show since 2005 has also seen improvements in SFX over the seasons, but they appear to have tried to bring ...