This is a UK question. Say I want to set up a website and sell
Twinings tea, can I just do that, or do I need Twinings permission?

Think of it as if you opened a physical shop (retail store)
- a big sign in the window saying 'Twinings Sold Here' is fine
- a shop name saying 'Twinings of Tooting' is asking for trouble

As long as you are not 'passing off' (pretending to be Twinings) you
should be fine.

www.FinestTeas.co.uk should be ... well Fine
( http://www.dnsstuff.com/ says it is not taken so go for it )

Maybe you should contact them and suggest the possiblity of setting up
an affiliate program.

That way you can sell their products on a commision basis and not have
to worry about setting up an infastructure.

I'm not sure if it would profitable, they don't have a strong online
presence for online shopping.

Do you think there's a market for this considering that you can buy
their products in every supermarket and a lot of cornerstores all over
the western world?