“We have nothing to lose but our aitches”

Solution 1:

This is a jocular reference to Marx and Engels' 1848 Communist Manifesto, which ends (in the 1888 translation by Samuel Moore):

Let the ruling classes tremble at a Communistic revolution. The proletarians have nothing to lose but their chains. They have a world to win.

Working Men of All Countries, Unite!

Although dropping initial [h]s has never been characteristic of all British non-standard dialects, from the late 19th century onward it was common 'eye dialect' in literary and stage representations (on both sides of the Atlantic) of lower-class British speech, a sort of metonymic emblem. For instance, alongside your Orwell quote you may set this from Lerner and Lowe's 1956 adaptation of Shaw's Pygmalion:

Hear them down in Soho Square
Dropping aitches everywhere
Speaking English any way they like.

The corresponding emblem for middle-class linguistic aspirations was the overcorrective intrusion of initial [h]; speakers were often depicted as employing both errors in the same utterance:

   “Hi’ve got henemies, miss,” he explained, “henemies has ’as sworn to ’ave my ’art’s blood, let alone rewenging my good name, and reporting on me at the station when Hi hain’t done nothink.” —Sala, Quite Alone, 1864

Solution 2:

My reading is that he is not talking about the pronunciation of the letter 'h' (correctly pronounced aitch) but the literal dropping (getting rid entirely) of the letter H in working class dialects.

So "have" becomes "'ave" and "his" becomes "'is"

the sentence: "Have you got his hat, Henry?" Becomes :"'ave you got 'is 'at, 'enry?"

The 'aitch' vs 'haitch' question: as I understand it, though pronunciation probably varies regionally, in general terms, the adding of h to aitch was the result of people being aware of the h dropping being considered working class and then incorrectly saying 'haitch' when trying to speak 'correctly'