The majority of people landing at Heathrow travel from there on London's very busy public transport. And yes, it is still busy. Look at recent pictures of the tube. Very crowded.
As for the original cases in Gloucestershire, of course in order for cases to be found, testing has to happen. I strongly suspect a lot of earlier cases were missed and the deaths (London and elsewhere in the UK) were mistook as heart attacks, strokes, flu, or pneumonia.
Residents in the south London boroughs where the SA strain has spread to, including the mayor of London, have spoken about delays and slowness of testing there.