The most important thing to remember is that layers are your friend!
The pictures above are me climbing in North Wales a the end of February and it was COLD.
Two pairs of normal black leggings from Sainsbury's (can't find them online but I'm pretty sure they'll be in any Sainsbury's that sells their TU range)
One jazzy pair of sports leggings from Decathlon - closest dupe I could find!
Sports bra from Nike. (bought from the outlet store at Resorts World, but look at this amazing sparkly one from ASOS!)
Thermal top from Marks & Spencer
Jumper (not seen) from & Other stories
A second jumper from H&M (bought in the sale so no longer available sorry! But H&M continue to have brilliant knitwear options that aren't too expensive.)
An incredible golden Primaloft Jacket from GAP of all places, bought in the sale for about £25 (I got extra discount because the waist elastic was a bit broken!) I feel like an astronaut every time I wear it. Other experienced climbers would probably recommend saving up for a RAB down jacket.
For the climb: It is VITAL you swap your cosy hat for a helmet (loaned from Warwick University Climbing Club). Better safe than sorry. Also swap your sturdy shoes for climbing ones - post here on a review of my beginner climbing shoes). And finally, put on your harness (this one was borrowed from WUCC too, but I have since purchased this one).
Hopefully this post shows you that you don't need to go out and spend all of your life savings in Go Outdoors for your first outdoor climb in the cold, you've probably got most of the necessary items. If you are going to buy anything new, make sure it's shoes! And a harness and helmet if you can't loan them!
Happy Climbing and hunt those bargains!