I arrive into Gatwick from Nice at 12:30 pm on 9 May, 2016 and depart from Heathrow at 15:15 for Shannon. What is the most efficient transportation between the two London airports? Will the airline transfer my luggage to my final destination on Shannon? NCE | LGW | SNN
Your best option would normally be the National Express bus that runs directly between Gatwick and Heathrow (Terminals 2/3, from where you can connect to the other terminals if needed). At that time of day it runs every 10-20 minutes, and takes a little over an hour.
Your baggage will not be transferred - you'll need to collect it at Gatwick, take it on the bus with you, and re-check it when you arrive at Heathrow.
However, 2 hours 45 mins is not a lot of time for that transfer. Allowing for getting to/from the bus that's at least 90 minutes, and you'll likely need to check your luggage at LHR at least an hour before your flight (airline dependent). That leaves you 15 minutes to get off the plane, go through immigration/customs and collect your bag.
Given that, I think your best option will be to take a taxi. It won't be cheap, but you don't really have time for any other option. Even with a taxi it's going to be tight...
The fastest form of transportation from Gatwick to Heathrow is undoubtedly a helicopter transfer.
HEATHROW TO GATWICK (or reverse)
UP TO 4 PASSENGERS | Flight Time: 00:25 (hh:mm) | Eurocopter EC120
£1,650 + VAT
Transit time should be 15-25 minutes.
If price is a factor then you might want to consider a transfer from a site nearby to each airport rather than the airport itself (saves on landing fees):
However using Redhill save a vast amount of money on Gatwick airports landing fee and is only 10 minutes by road from Gatwick.
London Heathrow Airport does charge a substantial landing fee, with this in mind we have secured a private take off site approx 10 minutes by road from Heathrow airport,
Since someone has already answered "what is the best", I will answer "what is the fastest". The answer used to be the Airlink helicopter transfer [15 minutes, £12 each way]. [Edit: apparently the answer is still the helicopter, but now the price is bigger.]
The fastest right now (without chartering a helicopter) is probably the Virgin Limobike. You won't be able to take much luggage (there is only a parcel box) and if you aren't used to riding on the back of a motorbike it might be uncomfortable. £150 each way, so also very expensive. And you need to pre-book, because they mostly operate in central London. But in heavy traffic, it's going to be the fastest.
The driver will provide you with a helmet and leathers, so you don't need to bring anything.
On FlyerTalk they compiled a complete list of different transport options, together with timings.