I don't go to the laundrette every week or so, but I do go when I want to wash big duvets and I went several times about 6 months ago when my washing machine broke.
Are you sure there is a laundrette within a reasonable distance of the bedsit? Laundrettes are quite rare nowadays.
Cost depends on the size of the wash. £3-£6 in the one near me, IIRC. The machines at laundrettes generally vary in size from the larger end of domestic washing machines to huge. King size duvets are no problem in the bigger ones. I'd guess 3-4 times the capacity of an average domestic washing machine.
You'd probably want to dry the washing at the laundrette too. Laundrettes usually if not always have dryers and the cost is usually very variable because you're essentially paying for drying time and it's usually available in a quite fine grading. The one near me divides drying time into 20p units. Probably a couple of pounds for drying a normal load.
It'll be up to the laundrette, but you're probably allowed to use your own washing powder, etc. They will probably also sell some, probably at a slightly higher price than normal. Which is fine, because you're paying for convenience.
It might well be a good idea to go to the laundrette you'll be using and see the costs and payment methods. They're usually coin-operated, usually unmanned and nearby shops are usually not very willing to be a non-profit changing service for the laundrette.