fanny is defined in Collins English Dictionary at CollinsDictionary.com as
(taboo, British) the female genitals
(mainly US & Canadian) the buttocks
It is used frequently to describe someone in a not very polite manner
You are a fanny! (often as complete and utter fanny)
'Taboo' seems a little strong for fanny as it can be used in a relatively friendly manner, particularly in the phrase fannying about. Collins notes that dick used to be 'taboo' but no longer is. Dick can also be used in the same way: dicking about.
Using fanny as a verb, still carries all the baggage and you need to be cautious with your usage.
Amongst friends stop fannying about means no more than stop messing about, however when used aggressively the meaning goes back to the original and becomes stop messing about, you fanny. Calling someone a fanny aggressively is only slightly less offensive than calling them a cunt.
You can use it self-referentially of course.
Sorry I'm late, I had to fanny about with my bike to get it started
but you wouldn't use that as an apology for arriving late for an interview.
Given the opportunity for someone to misunderstand the intention of the word fanny in fannying about I would avoid it in all situations where you are talking to people you do not know very well. It's generally OK with your friends (if they accept it) but it would not be the sort of phrase you'd use in a more formal setting.
Do bear in mind that both Dick and Fanny are still fairly common names in the UK, so if you hear someone using the word dick or fanny you need to comprehend what they mean. It's quite possible to say "I met a Dick/Fanny last night" meaning "I met someone, whose name is Dick/Fanny, last night" or it could mean "I met someone, who I really didn't like, last night"
Other 'messing aboutery' terms, listed in the order of what I consider least likely to offend to most likely to offend, the first three being not offensive at all.
- fooling about
- messing about
- mucking about
- bumming about[i]
- farting about (fart : passing wind from the anus)
- dicking about
- arsing about (arse : bottom/anus)
- fannying about
- fucking about
- cunting about[ii]
[i] bumming about is the US meaning of bum (vagrant, loafer, (UK:tramp)) and means much the same - 'hanging about doing nothing much'.
[ii] rarely used in a friendly way.