English speakers from the West Country seem to glottalize their tees just like Estuary English speakers do. I can't find a word about T-glottalization in the West Country accents on the internet. I'm guessing it must be a pretty recently acquired characteristic. Am I right? Is it spreading from around London?
T-glottalization is one of the features of the accent known as Estuary English. It was found first on the banks of the Thames at various points east of London. It has now spread more widely, so it’s not surprising to find it in the West Country as well. For more on the accent, see here.