In my days as a photo archivist, I spent a summer assisting a textile conservator to catalogue and re-house our collection of Civil-War era clothing - unfortunately, I've long since lost touch with her, or else I'd ask her expert advice!
A drycleaner would know how to actually store the quilt - it's a similar process to storing a wedding dress - but you'd have to be very careful to communicate that they NOT try and clean it, as the chemicals would destroy it. If you choose to do it yourself, the first thing you need to do is purchase some acid-free storage materials - I usually use Gaylord but I'm not sure they ship to the UK. *checks* It looks like they do, but it might be expensive!
Here's a list of materials you might use:
https://www.gaylordmart.com/listing.asp?H=3&PCI=128113Basically what you want to do is to avoid folding the quilt as much as possible - you're looking for a large, flat box rather than a smaller, taller one. You'll want to spread the quilt out and determine where you want to fold the material over (hint: NOT anywhere that it's been folded during the past ten years!), and then line the inside of the crease with bunched/rolled up tissue paper. This is really hard to describe - basically, you want to take a few sheets of tissue, gently bunch them in your hand so that you wind up with a springy loose cylinder of tissue, then place that on the quilt parallel to the edge you're folding over and then gently fold the edge/flap over so that the crease is rounded rather than flat. Does that make any sense?
Repeat the same process for any other folds/creases, making sure that you layer tissue in between every layer of fabric - you don't want the fabric panels to touch. Once you're done, lay the quilt in the box with more tissue so that it's completely separate to the sides of the box and hidden from view, then close up the box and keep it in a cool, dry area (I know, this is the UK, that might be a bit difficult - but under a bed should work pretty well). You can add a few packets of dessicant (silica gel) before closing up the box, if you want - Gaylord sell those as well.
Actually, they appear to have a quilt preservation kit here:
http://www.gaylordmart.com/adblock.asp?abid=8611&sid=3EAB897D62DD4AE2977794F7640365which has everything but the dessicant, as well as a guide (which probably explains the padding process much better than I just did). So I'd try that!