Learn English – ‘flip bits not burgers’ meaning

In the UK, certainly, a fanny is slang for vagina.

A lady's front bits would definitely mean vagina and not breasts. A less-oft-used term is "front bum" which is also slang for vagina.

I have never known front bits refer to breasts.

Merriam-Webster's Eleventh Collegiate Dictionary (2003) includes a definition of bit that matches the sense in which the characters in your example use the term:

bit n ... 3 : something small or unimportant of its kind: as ... c (1) : a small part usu. with spoken lines in a theatrical performance (2) : a usu. short theatrical routine {a corny comedy bit}

The relevant definition here is 3c(2).

Robert Chapman & Barbara Kipfer, Dictionary of American Slang, third edition (1995) has this take on the same meaning of bit:

bit ... 3 n fr[om] theater A display of pretended feeling, or an outright imitation; =ACT, SHTICK. [Examples:] So he does his hurt-puppy-dog bit/ You should see my Jimmy Cagney bit

An example of a bit would be someone doing an imitation of Edward G. Robinson as Johnny Rocco from the 1948 movie Key Largo—just a few lines of famous dialogue like

"After living in the USA for more than thirty-five years, they called me an undesirable alien. Me. Johnny Rocco. Like I was a dirty Red or something!"

That's a bit. So is a mime pretending to be trapped in a glass box. Or a magician doing the endless scarf trick. Or a comedian riffing on his mother-in-law.