Here's a list of all flights into Granada:
http://www.aena-aerop...ort-destinations.html
You'll see that (apart from Paris) there are no international services.
The nearest alternative airports are Almeria, Cordoba and Malaga.
The only service from the UK into Almeria is with easyJet, from Gatwick. Bus route 20 will take you into central Almeria but there are only 4 trains per day to Granada (with a journey time of around 2½ hours).
Cordoba currently only handles freight.
That just leaves Malaga.
easyJet, Aer Lingus, Monarch and BA all fly there from Gatwick.
CityFlyer Express operates from London City.
easyJet and Monarch fly from Luton.
easyJet & Ryanair fly from Stansted.
There are no flights from Heathrow.
The local train service from Terminal 3 will take you to Malaga train station. There are just 6 trains per day from the to Bobadilla (with a journey time of around 50 minutes) but just 3 trains per day from there to Granada (with a journey time of around 1½ hours).
However your salvation (unless you hire a car) would appear to be the direct bus service from Malaga Airport to Granada. There are departures at 1130 and 1830, with the journey taking 2½ hours for a 10 euro fare.
(Check the departure times, for your intended date of travel, here:
http://www.alsa.es/portal/site/Alsa )
If you're prepared to hire a car, these companies are at Almeria
http://www.aena-aerop...844290//Car-hire.html
with these at Malaga:
http://www.aena-aerop...947245//Car-hire.html
Chris