As FDos has pointed out you could take the train from T4 to Heathrow Central then change to the train to Terminal 5.
Another option is the free public buses numbers 490 and 482. They run from right outside the arrivals hall at terminal 4 and direct to T5 with no need to change (there are one or two stops along the way at some of the cargo centres). They run every ten minutes or so and take about 15mins. The journey is free.
MarkivJ, you should ask KLM if they are able to check-in your luggage through to Pisa from your departure airport even if your LHR/PSA flight is operated by BA.
Should it be the case, it will be possible for you to avoid going through customs when arriving at T4.
There is an airside bus connection between T5 and T4.
I used it once from T5 to T4 as I had no check-in luggage (BA doesn’t allow through check-in luggage with any other carrier when you have two different PNR).
So if KLM is able to check-in your luggage through to your final destination or if you don’t have any check-in luggage, you shouldbe able to have a quite smoothy airside transfer from T4 to T5 (based on my own experience of a T5 (BA) to T4 (AF) airside transfer without check-in luggage).
However please note that when I did it, I already had my boarding pass for my AF flight. So if you can’t get your LHR/PSA boarding pass either from your departure airport or from your own device, I have no clue …
It may be too late to be of use to you, but I landed today at 17.45 on a QR flight into T4 and, following the standard flight connection route, was in the T5 BA First lounge in 45 minutes flat.
Apart from the (extraordinarily) rude security lady in T5, it was a breeze.
