I have recently discovered that I can access FlightRadar's Filters simply by creating an account.
This lets me do things like display only one particular type of aircraft so I can see exactly how many Airbus A380s are currently 'in play' (in flight, recently landed or about to set off) as long as they have their transponders (transmitters, and receivers) on.
A380s are long-range aircraft so in the air for several hours. If I want to see one landing at Heathrow (LHR) I can either hover over likely candidates on FlightRadar's map and see where they're headed or tweak the filter to show only those landing at (or taking off) from LHR. If I wanted to visit Heathrow for a spot of plane spotting I'd easily be able to arrange timings so that I'm in the right place to see one coming in.
The Piaggio Avanti is a different and tricker beast to capture. As far as I can tell it's not involved in any regularly scheduled flights to or from London as they're all private aircraft. There will be some private individuals / owners being whisked (or whisking themselves) from some luxurious starting point to London, also some chartered flights where some people have booked City X to London and there are spare seats are available (for some eyewatering sums of money*, which is generally why I'm not on board as well).
The journeys are much shorter (also much louder) so I think to complete this quest I'd need to do take account of a few things, and I think luck will play a big part.
Biggin Hill: It takes me about 2 hours to get to Biggin Hill (a 90 minute bus journey but also have to be ready to leave the house and hopefully not have too long to wait for buses). Since many P-180 journeys are about this duration if one's due to land at Biggin I'd have to be on the move soon after it takes off (obviously it's better for me if it's coming from further away).
It also depends on the direction it's landing in as if it's coming in from the 'difficult to get to by public transport' part of the runway that's a non-starter (though I'd be able to check this before setting off, assuming the wind direction doesn't change while I'm en route).
City Airport: Despite living quite close to City Airport as the crow, or aircraft, flies, it's a bit fiddly to get to by public transport so I'd probably allow an hour or so. I'd take different journeys to get to different positions depending on which runway direction is in use. A P-180 coming over Connaught Bridge to land would be lovely.
So it looks like I probably need to check FlightRadar24 every half an hour every day to be reasonably certain of seeing one land! Alas... I am dipping in and out though. Just in case a likely candidate appears.
Current P-180 journeys at time of writing.
XGO2VE
Dortmund DTM departed 13:49, arriving at 14:20 at Braunschweig BWE (D-IINO)
OYO3
Pamplona PNA due to depart at 14:30, due to arrive at 16:09 at Paris LBG (F-HSIT)
OYO5A
Lyon LYN departed 14:09 local time, arriving 14:27 at Valence VAF (F-HXPG)
T7CHMG
Plovdiv PDV departed 15:10 local time, arriving Tbilisi (TBS) at 18:10 (T7-CHMG)
CGRTG
Detroit YIP departed 07:57 and no info about destination or journey time (C-GRTG)
N224JD
Chicago PWK 06:34 arriving at Nashville QQN at 07:51 (N224JD)
N413TB
St Louis SUS 06:12 arriving Tampa TPA at 09:28 (N413TB)
*Some prices have come down, when I first looked into it a flight was about £12k and some I've seen recently have been £6k. But still... it does seem like it would be a bit of an expensive indulgence!