I'm specifically talking about British English. In British English, "bath" (noun) has a long vowel ([ɑː]) while the verb "bathe" has a diphthong ([eɪ]) and sounds more like the American version of "bath".
Etymonline on "bath" says:
"bath (n.)
Old English bæð "an immersing of the body in water, mud, etc.," also "a quantity of water, etc., for bathing," from Proto-Germanic *badan (source also of Old Frisian beth, Old Saxon bath, Old Norse bað, Middle Dutch bat, German Bad), from PIE root *bhē- "to warm" + *-thuz, Germanic suffix indicating "act, process, condition" (as in birth, death). The etymological sense is of heating, not immersing."
And on "bathe":
"bathe (v.)
Old English baþian "to wash, lave, place in a bath, take a bath" (transitive and intransitive), from root of bath (q.v.), with different vowel sound due to i-mutation. Related: Bathed; bathing. Similar nouns in Old Norse baða, Old High German badon, German baden."
However it doesn't say anything about why "bath" has a long a vowel /bɑːθ/ and "bathe" has a vowel like American "bath" /beɪð/. Googling didn't help much. I also read different articles (like grammarly) but to no avail.
There is also a question in this platform but that does not explain this difference: "to bath" vs "to bathe"
Does anyone know the reason they are pronounced differently?
EDIT after the two answers:
I did not know the symbols that are used to represent the vowel sounds so I confused them.
I should not have compared the vowels in American "bath" and British "bathe". Also pointed out by the commenters.
Best Answer
I assume you're talking about Southern British English. Because 'bath' in Northern British English is pronounced /bæθ/, not /bɑ:θ/.
In Southern British English, 'bath' is pronounced /bɑ:θ/ and 'bathe' is pronounced /beɪð/.
I don't know what happened to these words in Old and Middle English but I'm going to apply some general pronuncation rules (Modern English).
The pronuncation of 'bath' was originally /bæθ/ in the South but due to a split known as trap bath split, the short vowel [æ] shifted to the long vowel [ɑː] before some consonants (/θ/, /sk/, /st/, /ft/ etc).
Examples:
According to Pronunciation Studio:
When we add the silent e to a word, it usually changes the vowel in a word to a diphthong or a long vowel.
Examples:
The /æ/ vowel seems to have a special relationship with /eɪ/.
When a word has the /æ/ vowel and we add the silent e to it, the /æ/ vowel often changes to /eɪ/.
This relationship can also be seen in words like profane and profanity.
The silent e often indicates a long vowel or a diphthong.
So the pronunciation of 'bath' was /bæθ/ and the silent e at the end of 'bathe' indicates that it has either a long vowel or a diphthong.
The Wikipedia article on silent e has also mentioned it:
The trap bath split did not affect 'bathe' so its pronuncation remained unaffected (i.e. /beɪð/).
It's worth noting that the vowel in 'bathe' is not the vowel in American English 'bath' as pointed out by Peter Shor in his comment.
The [ð] in 'bathe' is because of intervocalic fricative voicing.