I have noticed that in some British dialects, the long 'ee' sound is rendered as an'ay'. For example, 'me' as 'may', 'see' as 'say'. However, this is only an approximation as I am not sure how to transcribe it properly. I couldn't find anything about it on the internet. Could anyone tell me what this is called? (or if I am merely imagining it?)
In most accents of English, the words me and see contain the vowel sound found in the word fleece. This can be called "the fleece vowel" for short.
The fleece vowel is typically transcribed in the International Phonetic Alphabet (IPA) using a single vowel letter i (it can also be written iː; the symbol ː is a "length marker" indicating that the preceding vowel is long). However, it is known that in many accents of English, people actually pronounce this vowel sound as somewhat of a diphthong (a vowel that glides between different "vowel qualities").
Specifically, the fleece vowel may end in an "offglide" that is higher/closer than the rest of the vowel. This means that at the end of the vowel sound, the tongue moves slightly upwards towards the roof of the mouth. An alternative transcription that can be used to explicitly mark this gliding pronunciation of the fleece vowel would be ɪj, used by Geoff Lindsey in the blog post "The British English Vowel system".
It is notoriously difficult to distinguish the qualities of the vowels represented by the IPA symbols e and ɪ; thus, there isn't much difference between ɪj and ej. And ej is in the range of possible realizations of the "face" vowel (often transcribed as /eɪ/; Lindsey uses ɛj instead).
So even if a British English speaker maintains a psychological and phonetic distinction between the vowels of "fleece" and "face", a speaker of another accent of English with a less diphthongized "fleece" vowel might hear the British English speaker's diphthongal "fleece" vowel as the "face" vowel.