FD
I just use the underground all the way, as you say the Piccadilly line. If your case has a handle and wheels it is really no problem. On my recent trip I wondered if space would be a problem on a working day at 5pm but easy.
The only bad part of the trip is having to get the coach to Victoria and change there but it also goes to Sunderland, North Shields, Whitley Bay, Blyth, Ashington on the return.
I swore I would never use KLM again after they laughed in my face over the over booking episode. That was when I used KLM long haul and was already in transit and even though I had a reconfirmed and paid for seat they still said, "sorry sir, the plane is full and you're not on it". As far as I'm concerned a plane seat is no different to a car or a TV set. If I have paid for it it is mine and no-one has the right to sell it to someone else.