I think you have this slightly muddled. It's not that upper-class Brits
  pronounce more of the "h" sounds. The idea is that upper-class types go
  to private schools and they learn French, and so they know that lots of
  English words that have a leading "h" are French loanwords and thus the
  they are unaspirated in the original, i.e. the "h" is silent – it ought
  not to be said. "Hotel", "hospital", "honour", "homage", and so on - no
  leading "huh" sound. Which means that the word starts on a vowel sound,
  and in turn, that means it takes "an" in front not "a". 

  But people who don't know French and don't know which words have a sort
  of silent H on the front and which don't try to hyper-correct – so they
  start inserting extra initial "huh" sounds on words beginning with some
  form of vowel.

  This is how old British TV comedies tried to parody policemen talking -
  they are often working class but in dealing with lawyers, judges, etc.,
  they are aware of speaking to someone from a higher social class and so
  they try to speak more "proper" and insert loads of extra "h" sounds.

  "Hin my hopinion, your onner, the suspect was hup to something."

  Drop the "h" sounds where they _should_ be, and at the same time insert
  extra ones where they shouldn't be.

  I think this is what the Pythons were playing with. Now that Britain is
  a little bit less class-conscious – and foreign accents are heard a lot
  more often – it doesn't seem funny any more so it's gone away. That's a
  good thing overall, I reckon.

  (Wondering how long I should keep this up...)

Forever, I'm finding it very entertaining

  > Forever, I'm finding it very entertaining

  Why thank you! :-)

  There's reams of it on my late-1990s homepage, which I resurrected and
  uploaded a few years ago. I did an extremely cursory update but expect
  plenty of broken links and things.

  Finding it is an exercise for the reader... ;-)

Expensively educated people might pronounce French loanwords by dropping the h in an ostentatiously French way (and maybe making homage rhyme with fromage) to showcase their etymology knowledge and are more likely to use the grammatical convention[1] of prefixing them with 'an' rather than 'a' even even when sounding the 'h'. But the "correct", RP way their expensive education teaches them to enunciate the words you listed as English words is with a leading h and has been for a century or so, except 'honour' (also 'honest', 'heir'), which pretty much no native speaker pronounces the 'h' in.

[1]now usually an historic grammatical convention ;-) it survived longer than the silent 'h' in 'historic' in conventional English speech though...

On the other hand a leading 'h' is routinely dropped in spoken English in most words in many British dialects spoken by less expensively educated people (especially southern urban "working class" people), which has no more to do with French etymology than the glottal stop that means some of the same accents also drop a lot of 't's from the middle of words. That's where all the twentieth century jokes involving Cockney stereotypes overcorrecting by inappropriately adding 'h' when trying to speak 'proper' come from. The overcorrecting in turn came from "upper class" people being snobbish about the sort of person who 'abitually indulged in "h-dropping"...

As for the name of the letter, according to the OED and RP dialect the sort of expensive schools that emphasize correct pronunciation teach, it's is definitely called 'aitch' in British English. And has been for several hundred years. Not all people who pronounce it 'aitch' are "upper class" (there's a rapper with the stage name Aitch) but all people who are "upper class" are taught to pronounce it 'aitch'. On the other hand, it's a bit weird that the name of the letter doesn't contain the letter, so lots of British people whose education doesn't involve teachers emphasising The Correct Way To Speak feel the name for the letter 'h' should start with the letter 'h', hence 'haitch'.

Also, Irish dialects call it 'haitch'; in Northern Ireland it's apparently a shibboleth distinguishing between Irish-identifying Catholics ('haitch') and British-identifying Protestants ('aitch'). Suspect this also has nothing to do with the French :)

So "upper class" learn to enunciate the letter 'h' without an 'h' in it, but are taught to pronounce it more or less everywhere else (with a few quirks to keep them honest) and others more likely to drop the 'h' sound from other words but add it to the letter name. Yes, English pronunication is weird

For those desperate for even more 'h' discourse, seee: https://www.oed.com/dictionary/h_n and https://en.wikipedia.org/wiki/H-dropping

> Also, Irish dialects call it 'haitch'; in Northern Ireland it's apparently a shibboleth distinguishing between Irish-identifying Catholics ('haitch') and British-identifying Protestants ('aitch'). Suspect this also has nothing to do with the French :)

Fascinating. I didn't know that, despite my Belfast-born dad being the reason I'm a citizen.

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

https://www.bbc.co.uk/programmes/p00hhrwl