Flights

That is a small airport not many gates; additional flights are seasonal and October is off peak. Even Chicago has airlines that fly places seasonally.
Do not waste your time reading reviews all airlines have them. I have flown Vueling 3 times and the flight landed fine no safety issues, no one had an attitude or canceled the flight. What is a good airline to you? I can probably send you some bad reviews about it. Vueling is a subsidiary of IG Group which owns Iberia British and Vueling

Does it have to be Gatwick or will any other London airport do? Bologna is only 80 km from Florence and has direct flights to LGW with Easyjet.

Easyjet also has flights from Pisa.Direct regional train to Pisa Centrale then change to the Airport People Mover. Takes about 1 1/2 hours.

Airlines released their October schedules some time ago, so it’s unlikely that any further flights will be released.

Vueling are a subsidiary of IAG, who also own British Airways and Iberia. They aren’t a subsidiary of BA. Vueling are ok - like all discount airlines - know the rules on luggage and ensure that you comply. I wouldn’t bother taking a train to another city to avoid using Vueling.

Is there a reason for Gatwick? Vueling is OK. They take off and land on time and that is about all you can ask for in a discount airline. Second, Read and fully understand their rules for luggage, check-in, etc. These airlines take in huge additional revenue from folks who think that an extra two inches or five pounds, etc., will slip by them. Nothing slips by because it is all a revenue source. easyJet is our preferred discount airline but don't know if they fly out of Florence without checking schedules. Knowing a bit more what your travel plans might improve our responses.

It does appear there is just the one direct flight from Florence to LGW.

If you are flexible about which airport you fly from, (Florence, Pisa Bologna), and where in London you land the choice expands significantly.

At this point there is no reason to expect any flights to be added and as time goes by the planes will fill and price will climb

Ryanair is the biggest carrier out of Pisa International. We flew from Gatwick down to Pisa on EasyJet.com without incident--2 hr. flight. Heathrow is more of an international airport--gateway airport to all over the world. Gatwick is more of an airport with connecting flights to all over Europe., and some long haul flights into there.
Many complain about Vueling, but all the budget airlines get complaints. We've had good luck with EasyJet over the years.

Does it have to be Gatwick? British Airways has a flight into London City and it isn't much more than the Vueling flight into Gatwick.

London City is a small airport in London itself. I have flown in and out numerous times and would prefer it over Gatwick.

From Pisa, a bigger airport than Florence, you have three Easy Jet flights into Gatwick. I have flown Easy Jet numerous times including three times in the last month. No problems. Although for longer flights, I do suggest paying a little extra for the extra leg room seats. Two of my flights were over three hours and I'm glad I did.

No matter what airline you choose, read their rules carefully. If you don't, it could cost you.