I can help.
You'll need to register it as a new society, I struggled to find any guide on the lancaster union website (it would be classified as a recreational sports club). They do have a terms of reference for their societies committee which goes:
5.8. New societies will be approved based upon the following criteria:
5.8.1. Activity is in line with and supportive of the Students? Union?s charitable objectives and strategic vision of ?inspiring and supporting students to love Lancaster?
5.8.2. Society must not duplicate an existing opportunity and there must be a proven interest from the student community in the activity the society intends to provide: there must be a minimum of 20 expressions of interest in membership
5.8.3. The first year of the society must be budgeted and demonstrate that activity is financially sustainable
5.8.4. The executive must have researched and evidenced that the Union and University has the facilities to accommodate the activity this society intends to provide; or have identified an alternative solution
5.8.5. The proposed executive must demonstrate a competency and understanding with regards to fulfilling the executive roles of president, treasurer and secretary
This is an unusually high standard from what I've come across in the past.
5.8.1. sounds fantastically Orwellian.
5.8.2. 20 people is quite a lot - it may be that the idea of expanding the mountaineering club's remit to include occasional caving and then forming a society when there are enough people involved/sticking with the club as they might afford financial security, easy advertising/manpower etc.
5.8.3. Is surprisingly easy when you don't have any money - What we'll need to do for this is either sort out a lending of equipment (there are numerous clubs that could help with this) long term or do a commercial trip to trip lending fee which will factor into the cost of each trip. A budget for next year would have to be submitted I'd have thought late march/early april - that would include membership fees and the expected revenue 20x15=?300 which won't get you very far at all, then you'll make the case for getting money out of the safety budget - lights/helmets/wellies/oversuits (I'll leave SRT out for now). When asked what the club is providing for that fee the answer will be insurance (?11) and the rest to cover organizational costs.
5.8.4. This is really about storage space for kit: i.e. ask if they have any and when they say no (that should be if) then it could be housed at a member's house - or in the shop you're renting from if you haven't got that far.
5.8.5. This will be achieved by submitting a copy of a workable (and preferably pretty lean) constitution. Competency of treasurer would be given with a budget (given how many student clubs submit budgets) but if not then it might be an idea to get people who've served on other committees to at least nominally be a part of your one.
Message me and you can have even more dreary tips and details. I can't emphasize enough that though there's a lot of paperwork, chances are I've probably already done it for you