I support SEP Flyer in that Fairoaks makes a good potential option if located in Vauxhall.
I have been regularly flying a 172 in and out of Fairoaks for maintenance for several years. Consequently I have frequently used the trains and taxies to get there to collect it and get home after dropping it off. I’ve always found taxies at the station. Typically I’ve not waited more than 10 to 15 mins for a cab to come and collect me when I’ve telephoned from the airfield to go to the station.
Woking station is in fact a direct service from Vauxhall train station (44 min according to National Rail and quicker if you change at Clapham Junction to a fast train) and the taxi is about 10 mins.
From my experience of travelling around London distance is not a good metric for comparing between different venue locations. When travelling in London you also need to consider the route and types of transport you want to use and the impact of congestion on your journey. Time of day really does make a major difference to journey times; also, whether it’s a weekday or weekend when you need to do it.
I’m with John R81 - visit the airfields around London and choose one where you feel at home and that has road and public transport links that are good for the times you will need to use them.