The BBC recommends the following minimum bandwidths for iPlayer content:
HD TV: 3.5 Mbps
TV: 1.5 Mbps (for the highest quality service - lower bandwidth options only require 0.5 Mbps)
Radio: 0.128 Mbps.
YouView recommends a minimum speed of 3 Mbps for its service.
Unless you're into online gaming against other players (where a mouse-click from you requires an immediate response on your opponent's screen) I can think of no reason why your internet speed should be regarded as inadequate.