It might be cheaper to use a very inexpensive bag - maybe one from a charity shop and leave it behind at your first accommodation, then buy another cheap bag when you are ready to go home.
I was going to suggest the IKEA Frakta bag with zipper, but it looks like they aren't available in the UK anymore. However this IKEA bag should work.
KNALLA bag, black/white, 40x25x47 cm/47 l This bag is just as practical for weekly grocery shopping as for dirty laundry or beverage cans. And best of all – you can use it again and again. Perfect for transporting most things, from laundry and groceries to recyclable items.
www.ikea.com
Or this one for £2 more
GÖRSNYGG bag, blue, 40x30x60 cm/72 l This durable bag holds a lot when you´re on the go and takes up little space when it´s empty. A real everyday hero in blue with striped handles that are just as easy to carry in your hand as on your back. Easy to keep clean by just wiping with a damp cloth.
www.ikea.com