Martin, as far as I know, there are none close to Bank. If you go sarf the closest may be Tackle & Bait on Nunhead Lane. There's also Swifty's in Vauxhall. (Wilson St. I think) If you go north the only one I know is Ashpoles on Green Lanes at Newington Green. There is definately a lack of tackle shops in Central London. Decathlon at Surrey Quays has a limited fishing section. Lots of French stuff. All seems a bit foriegn to me.
Oh, and of course there's Farlows of Pall Mall. Mainly game fishing.
Hope this helps.