Hi,
If you have no unduly large area of carpeting, bare floors and no pets about you should look for either a small canister vacuum or what we call call a stick vac here in the U.S. They may be your best practical choice. A small canister vac will allow you to do rugs and floors plus let you go over furniture and woodwork if you need to. Also a small uncomplicated machine should be less problem to store when not in use. Do not buy a rechargeable vacuum but one that plugs into an electrical outlet for continuous use. Rechargeable vacs are meant for quick pick ups and have short effective run times. They best serve in households that already have a standard vacuum cleaner.
Forget about Dyson as it prices ridiculously. Vax, Hoover, Morphy Richards, and Russell Hobb I am told usually have a decent supply of affordable and simple machines on the market in the UK.
The choice of a bagless vacuum as opposed to a bagged vacuum means trade-offs either way. Throwing away a vacuum's disposable bag is usually neater and more convenient but you must purchase bags and filters as needed. Bagless vacs -- any brand -- can be messy to empty and call for regular checks and maintenance re their filters to keep them running well. However, with proper care, the few minutes you'll lose to do so will spare you having to buy replacement dust bags. If you decide on a bagless vac be sure to learn if all filters are washable as buying replacement filters can be pricey.
If you're really looking to save, try visiting a local vacuum sales and repair shop in your area that has used and refurbished machines up for sale. Make it plain that you are not out to buy the moon and have them show what they may have on hand in that category.
Don't expect a lot if you haven't much to spend but, used or new, test for yourself whether noise levels, size and weight are acceptable to you.
Whatever machine you get, be sure empty it regularly -- bagless, after each use and bagged at around three-fourths full -- AND read the machine's user guide so that you can use it properly. You should have little problem with your choice if you follow those simple steps.
For starters, check out -- http://www.argos.co.uk/static/Browse/ID72/14418639/c_1/1%7Ccategory_root%7CKitchen+and+laundry%7C14418476/c_2/2%7C14418476%7CVacuum+cleaners%7C14418639/pp/Show+all/r_001/4%7CPrice%7C60+%3C%3D++%3C%3D+99.99%7C2/r_003/5%7CPrice%7C60+%3C%3D++%3C%3D+79.99%7C2/s/Price%3A+Low+-+High.htm
Good luck,
Venson Thomas