Only one thing to remember you will need customs and emigration in and out at which ever you plan to fly into, so will have to call ahead with flight details and passport information, flight plan does not count. This may need to be done 24 hours ahead as a minimum. No problem if EU citizens more difficult if not. If an airfield has it on site can make it easier.
Just a thought. All the fields ringed around London can take a PA28 and most are within 5/8 m/k of a train station with 30 to 45 mins into London.
So big choice its down to final destination in London and what train line ends up closest to your destination.