Strange you ask a question to which you already know the answer. Yes, go ahead and learn on the Tiger!
If money and time had not been an issue I'd have learned on a taildragger. As it was I went from gliding to Robin's and then, when the club moved away, finished my training on Cessnas. It was many years later that I actually paid to fly a Cessna or Piper aircraft again.
My point of view is that the skills you will learn from the Tiger Moth are the essentials (although the purist in me says you should go solo in a glider first!) and will be well learned.
I don't know any of the instructors at Cambridge but would be interested to know their statistics re. hours to completion of course for the Tiger versus Cessna students. There is no reason why it should take any longer.
Go on, you know you want to!