The Piccadilly line will be standing room only from about 1530, you will have no personal space.
FWIW, the Piccadilly Line trains tend to be less busy towards the back of the train. It's the wrong end of the train for getting out at T5, but IMHO the extra 90 seconds walk along the platform when you get to T5 is well worth it if it means you get a seat for most of the journey from central London.