In this context. 'going' could probably be expanded to 'going round and round in the washing machine*', but as others have said it the general meaning is that the process is in train ''going on".
b
PS * This may attract accusations of cultural bias, but the person doing the washing is unlikely to forget about the need for drying while doing a hand-wash.
PPS Perhaps the trouble with understanding this was the word laundry. There may be dictionaries that give the principal meaning as a place. (In fact when I was working at OUP I lived in a building that had previously been an out-house on an estate (mentioned in an oft-quoted poem by someone beginning with G, but that would be taking the tangent a bit too far... ;-)) - and the house was called 'The Old Laundry'.) The expression 'I had laundry going' uses 'laundry' to refer to the things being washed.