The short answer is that the answer is not easily found on the internet. The Gloucestershire Echo did a detailed five-part series on this in 2015, but it is no longer on their website. So we are using offline methods to answer this question and will report back shortly.
There are many parts to this answer: the most obvious are (1) why did GCHQ feel the need to move and (2), having decided to move, how did they choose their new location? It is the second which is more interesting and more difficult to answer.
The first question is easily answered: anyone who has visited Bletchley Park, now a popular museum, will recognise that it is not fit for a modern computer-heavy organisation. More than any other branch of intelligence, and since it was created in 1914, GCHQ has needed ever more processing power. Even today, computers on this scale take up a lot of space, and a secret organisation cannot rely on public ‘cloud’ data servers. They need their own, and these are more secure and more effective if they are on-site.
So GCHQ left Bletchley to Eastcote, and quickly outgrew the Eastcote site too. One suspects that there was a strong pull towards London, where MI5 and MI6 have always lived. But unlike MI5 and MI6, GCHQ is not primarily focussed on HUMINT or human intelligence. Proximity is less of an advantage and, some believe, actually a disadvantage.
Cheltenham puts GCHQ closer to America, but this is likely a red herring. It is on cheaper land, in an area with a strong military history. It can quietly sieve through information collected from all corners of the world in splendid isolation. Out of sight, out of mind. Until, that is, Geoffrey Prime came along, a mole within GCHQ, discovered in 1982, yet active throughout the 1960s and 1970s and convicted in a trial. For some reason, perhaps because SIGINT is less glamorous, Prime never achieved the notoriety of the Cambridge Five.
In the early 2000s, GCHQ moved into its current ‘donut’ office in Cheltenham, however it also has outposts in Bude, Cornwall and Scarborough, North Yorkshire. I discovered a range of large receiving dishes near St. David’s in Wales, but these are not publicly accepted to belong to GCHQ.