It does indeed in the UK and perhaps in some other countries. Not in the US, where it has the meaning Michael describes (which is not used in the UK), and where the other activity would be called "dish washing".

It depends on what variety of English you are using. GB English: 'Wash up' means to wash dishes. US English: 'Wash up' means to wash your face and hands, for example, and tidy up your appearance in general.

'wash up' >clean cookery or wash hands or face

Usually to "wash up" means to clean yourself to some extent. Such as "Give me 5 minutes to wash up before dinner" meaning that you are going to wash your hands and perhaps change clothes.