Easy to say it on here, but I suspect the reality is VERY different.
1/ Cost; it would be ridiculously expensive.
2/ Are there planes and crews available?
3/ Are there flight paths available?
4/ Can the airports at either end cope with the extra flights?
Having worked in or around the Aviation Industry for 40 years......
1) London City airport is mandated closed from Midday Saturday to Midday Sunday. This was required to get planning permission to build the airport.
2) The largest aircraft allowed (Embraer 195E2) in London City has a maximum of 124 seats. So you would need between 5 and 6 of these aircraft to replace every train
3) Airports such as Gatwick/Heathrow/Luton & Stansted in London, Charles De Gaulle & Orly in Paris, Schiphol in Amsterdam and National in Brussels are all slot restricted and generally operate at close to full capacity, with almost no extra slots available.
4) If you could find the aircraft, then flights from Cambridge or Southend to Rotterdam/Charleroi/Beauvais/Liile would potentially be possible, as long as all these airports had the staff available.
5) I could source an A320 (150 seats) at 4 hours, and another within 12 hours, but getting all the aircraft needed would take months of planning. Most availabl aircraft will be in storage in either Spain or the Baltic States where most of the ACMI (Aircraft, Crew, Maintenance & Insurance) carriers are located. For example planning potential aircraft capacity for the Champions League final in Wembley in June actually started earlier this year, and obviously we have no idea who the finalists will be!
6) For a single 150-200 seat aircraft you will need 2 pilots, 4 cabin crew, 1 dispatcher, 3 baggage handlers, 2 checkin agents, and 2 gate agents. Obviously you can re-use the ground crews on subsequent departures, but you also need the same numbers at the other end, plus aircraft cleaners.
7) If you sourced an aircraft for a 10AM flight, then it wouldn't be available for the next flight until 1:30PM. Think 30 minutes boarding, 45 minutes flying to Paris, 20 minutes deboarding, 10 minutes cleaning, 30 minutes boarding, 45 minutes flying back to the UK, 20 minutes deboarding and 10 minutes cleaning. An hours turn round is reasonable for a smaller aircraft, but it gets longer when you use bigger aircraft.
8) For one round trip UK to France at short notice, you wouldn't get change out of $30000.