The first time I ever went into LHR was in a Piper Arrow,
We had to have a handling agent (we selected British Airways) who arranged all the paperwork and gave us one of their landing slots.
Handling agent fees came to around £50 and the landing fee (outside peak period) was around £98. I believe the peak period fees would have been about £300.
ATC kept us at 6,000 downwind - asked for 180 knots to the outer marker (didn't get it). We advised them that if we could "land long" we could keep the speed up. We stayed "clean" and flew, with power on, to the threshold in a dive that gave us about 140 knots - flared over the numbers, floated along the runway and selected gear down at Vle. Worked a treat.
ATC were brilliant - ground movements controller really helpful...and ground movement was the hardest part (tricky taxiiways for a fairly inexperienced PPL).
I'm told, now, that single engine aircraft are banned from use of LHR due to rule 5. If you had an engine failure on approach to the westerly runways, you could not glide clear of the built up areas.
Been back many times since - in commercial aircraft...but always remember that first visit.
Gatwick, of course, will let you in even now!! Just as much fun - and just as much money!!