Thameslink train services providing direct services between Great Northern stations and Gatwick Airport via Finsbury Park are due to start in May, but as of 26th February a "preview" service has been running..
The service is very limited - three trains a day in each direction on Mondays to Fridays only, all between the morning and evening peaks. They run between Peterborough in the north and Brighton or Horsham in the south, but of interest to Palmers Green people is the fact that they all serve Finsbury Park.
Southbound trains run from Finsbury Park through a new tunnel to the Thameslink part of St Pancras International (they don't call at Kings Cross) and thence via Farringdon, City Thameslink and Blackfriars to London Bridge. They then continue via East Croydon and Gatwick Airport to their ultimate destinations.
These initial services have very long amounts of "padding" built into the timetable (eg a 17 minute wait at London Bridge), so won't be speeding up anyone's journeys.
You can read about the first day of operation on the Diamond Geezer and 1London blogs.