Not at all - but note that there are three tube stops at LHR (T1-3, T4, T5) and that not all trains go to T5 - check before boarding (although if it's a quiet T4 train, get on anyway and change at Hatton Cross...) There are also some Piccadilly Line trains that don't go to Heathrow - don't get on these!

You can walk (via a couple of travelators) from T3 to the Tube - it's a very well signed underground passageway. The T5 Tube is pretty much directly under the terminal.

I'd also add, for both Tower Hill and Westminster, don't change at South Kensington, but rather at Hammersmith or Barons Court, where you just have to walk across the platform rather than up/down stairs and escalators.