Anthony Burgess discusses this in A Mouthful of Air, his book about the English language. I don’t have a copy to hand but, from memory. Cockney or Kentish speech (I forget which) at the time featured an intermediate sound, between the ‘v’ and ‘w’ that are familar to us, and this is what Dickens is pointing to.
You make a ‘v’ sound by bringing your lower lip against your upper teeth, and a ‘w’ by putting your upper and lower lips close together, but without involving your teeth. Now try to make a ‘v’, but configuring your mouth as for a ‘w’. You’ll find it very difficult, your lips will come closer together than they do for ‘w’ and there’ll be a more forceful exhalation of breath. That’s, more or less, the sound that to which Dickens is pointing. It no longer features in Cockney/Kentish speech.
Why would he have English characters use a Hawaiian pronunciation? At any rate, no, he didn’t, but since w is pronounced as v in German and other European languages he would have hardly have had to go as far afield as that. (But it’s clear that this is a homegrown English dialect and not any foreign influence.)
The sound in between [v] and [w] that you’re describing is the labiodental approximant [ʋ]. I’m familiar with it from Hindi/Urdu and Indian English accents. Sometimes it really does sound as though /v/ and /w/ have been interchanged (like in the example “wery vell”), but I wonder if that’s an effect produced in the brain by hearing [ʋ]. Where /v/ is expected, it sounds wrong, and the brain slots it into the /w/ phoneme—and wice wersa. That’s what phonemes are, they’re slots in the brain’s phonetic map, and each language has only a certain set of them. Phonetic sounds from outside those slots are likely to be slotted into the nearest phoneme. Since [ʋ] is midway between /v/ and /w/, when it sounds off where one phoneme is expected, the brain slots it into the other one. What I’m saying is maybe the interchange is only happening in the brain. If someone says /ˈʋɛɹi ʋɛl/ with the same initial sound in both words, the brain can interpret that as “wery vell.”
Out of curiosity, what is the difference? The dictionaries with IPA aren’t helping me out, as they give /v/ as the starting consonant for both. Is one more between a /v/ and /w/, for lack of better description (labiodental approximant?) Is this standard in all varieties of German, or is it dialectal?