It was redesigned because someone nicked the original when it was on tour. Though when I visited it, I didn't think it was the original, but more mock-up done by someone who had glanced at it once.
I do remember Ian & Barbara being suspicious of Susan's answers and wanting to meet her Grandfather and following her.
Even as a small kid that seemed like a couple of massive plot holes, I may have missed part that made it more believable:
1) Why was she at School /College?
2) How did she get enrolled without address, biirth cert etc?
Someone once said the Best SF is where there is only one "unbelievable" fact . Of course Dr. Who isn't SF now, if it was once. Sort of a rubbish mixture of Horror and Teenage drama now. I guess it gets such good ratings because the dilution of Multichannel T, Reality TV, Soaps, phone in and such that TV is generally now so poor we are reduced to to watch DVD box sets.
If there are repeats of any good programs they are either on daytime or pay tv.