I always take the train when someone else pays. Also sometimes when I’m paying, but I have to book well in advance and get the cheap fares etc.
I put my bike on the train to but that’s made a bit harder now you need a reservation. But thne I can ride down the station and ride around in London when I get there. But then again I generally know where I am going.
If I had to drive, I’d keep on the M4 until it ends then keep going straight on until I get to Hammersmith, then park at the multi-storey above Hammersmith tube. Not the nicest approach, but it’s convenient.
Also parked in Hayes before. Come to think of it, you could probably park at Heathrow too.