Galway to London?

You either go down to Shannon Airport on Bus Eireann Route 51- flights to London Stansted or Gatwick with Ryanair or to Heathrow on Aer Lingus,
or catch the aircoach to Dublin Airport which has a wider range of departures to all London Airports

Where are you staying in London?

Tower Hill

I wanted to see if any airport had time benefits from where you were staying but it will take about the same from any London airport with direct flights if you fly from Shannon. Plan an hour to an hour and 15 minutes.

If you decide to fly out of Dublin, you can get a direct flight to London City Airport which is only 30 minutes to Tower Hill.

Note that you can also now fly on Aer Lingus from Ireland West Airport Knock to Heathrow - you get there from Galway on Bus 64