As a UK citizen, do you need a passport to fly from London Gatwick to Edinburgh with easyJet or British Airways? And also will this be the same at other airports?
The short answer is no, and this answer provides good references with more details.
In answer to the supplementary question that you raised in the comments section, airport security do not require any ID, so they will not be asking for a passport either.
In the case of Gatwick, if your flight isn't leaving the UK, they will take your picture as you enter security, and check this before you reach the gate to prevent an immigration scam that became quite prevalent a few years ago.
I can't find a reference to this online, but this is from extensive personal experience. I would have added this as a comment, but apparently my reputation hasn't preceded me :)
Some airlines accept photo driving licences and other forms of ID for domestic flights, but many don’t so check your airlines website when you book your flights. If you turn up with the wrong type of ID you will not be able to fly and you are unlikely to get your money back.
Source: UK Civil Aviation Authority
If you are flying solely within the UK, including Northern Ireland, you do not need a passport but we advise that you carry photographic identification with you when travelling, such as your passport or driving licence. This may be requested at certain points in your journey. Children under the age of 16 years do not require identification to travel within the UK.
easyJet doesn't require it either.
As an EU citizen I once tried to travel intra-UK with only my UK driving license, which the check-in clerk did not want to accept at first (they told me that a driving license is only valid if I'm a UK citizen), and it took around 10-15 minutes of arguing, showing them their website on approved IDs, and them calling up their manager so they could finally accept it.
While this does show that you don't need a passport, please note that it did delay the check-in process, meaning if you're in a hurry it might still be beneficial to provide a passport, especially if you're not a UK citizen.
At Luton airport a few years ago a police officer stopped me and asked to see my passport. I said "I don't think I need it", he said "yes you do", but when I mentioned I was flying to Glasgow he let me pass. (Why exactly there were police stopping random people is something I don't know, nor what good it does if anyone can just say they are going to take a domestic flight.)
As it happens I did have my passport with me just in case...