Do you know if you want a neutral or a supportive sole?
It might not matter so much at the outset, but might become more important as you go on.
To find price reductions, you need to look at discount sites for discontinued lines and previous year's colours.
I'm finding it hard to bring in trainers I know are good for both beginner runners and for general gym wear within your budget. The ones I would recommend (both neutral sole, used by me or DD for running and general sport) are the Nike Vomero and the Asics Dynaflyte, but I've yet to find a pair under £60.
So other places to try are Decathlon on the high street (staff may or may not know their stuff, but I find the chances of getting someone who does are rather higher than in other sportsy chains) who seem to have lots of their own brand Kalenji well within your price range. I've never had a pair of these, but I buy lots of other Kalenji stuff and found it good.
Or try Wiggle - a fairly serious triathlete's site, so I suspect everything they stock will perform well.
www.wiggle.co.uk/womens/run/running-shoes/?o=2
I'd try either Asics or New Balance first from the brands they have discounted at the moment.
Remember to size up a size from normal shoe size when getting running trainers