Just to elaborate on the Piccadilly line; it's a normal subway train (part of the 'Tube' that runs from Heathrow west then north-west through the centre of the city). The train-cars are specifically designed for extra luggage as it's always been the 'airport train' so areas around the doors in the centre of each carriage have less built-in seats, instead offering a larger wider luggage area with folding seats for when this space isn't used by luggage. These trains start their journey at the airport pretty much empty so you shouldn't have a problem getting your luggage on board or getting a seat.
However - if you need to change to another line when you get into the centre - especially if this is during the morning or evening rush hour then the prospect of continuing your journey with big luggage gets a bit more complicated. Not insurmountable but definitely not as easy or pleasant. My advice if you have to travel between 7-10hrs am or 17-19hrs pm is work out the nearest stop to where you'll need to be on the using only the Picadillly line, then come out of the station and hail a cab for the last bit of your journey.