Really, you cannot classify British humour as anything. Most people associate "Monty Python" as being "British", but its leaps away from something like "The Office". Even "Fawlty Towers" co-written by a Python has little in common with "Monty Python".
Wallace and Gromit's humour is similar to that of "
Last of the Summer Wine", which actually stars the voice of Wallace, and it also the longest running British sit-com. So I guess "Last of the Summer Wine" can be described as typically "British". But Last of the Summer Wine is written by the same person as "Keeping Up Appreances".
I haven't seen the new movie, but based on the shorts, I would say Wallace and Gromit do share traits with "Mr Bean" (lots of visual, non-dialogue humour) and I've heard the new film has a lot of innuendo (meaning it will have some traits with "Are You Being Served?" - although the average American probably never uses the word "Totty", and some of the other innuendo if they use words like "Crumpet", so this may go over some people's heads).
The short's humour seems to be that of 70s Britain without the racism and sexism which was surprisingly common at the time. Wordplay and puns, visual stunts and odd surrealism. Like the characters of the movie, Wallace and Gromit are of a Britain gone by.
British humour now seems to be a reaction to all the American sit-coms screen over here full of family values and morals. Therefore, homegrown British comedy is often dark, black humour or way out surrealism. See
"Nighty Night" for an example of the former,
"Father Ted" for an example of the latter, and
"The League Of Gentlemen" for an example of both combined.