How does the pronunciation of 'the' vary in British English?

To cut through the comment chaos and address the question:

You can say "the" like "thee" or "thuh", interchangeably. This isn't a recent development and as far as I know it's not particularly associated with a particular region. A given individual might use both on separate occasions.

I believe that "thee" is favoured in "Received pronunciation", which is a now largely discredited guide to how English is "supposed to be pronounced", derived from listening to how the royal family speaks. That is to say, the Queen would most likely say "thee". Even the BBC (a good touchstone for any British English pronunciation questions) don't insist that their presenters speak like this, however, any more.

https://en.wikipedia.org/wiki/Received_Pronunciation

Choice of "thee" or "thuh" generally depends on the word that follows. Words starting with a vowel sound will be more likely to have "thee" before them, as it's an easier transition. For example "Thuh team" is easier to say than "thee team" and "thee orange" is easier to say (or perhaps easier to hear clearly) than "thuh orange", which would sound too much like "thorange". Hence, people will tend to say "Thee E.U.", as it's just more natural to say than "thuh E.U."