Yes cheap is relative, obviously you can pop to your local registery office with just a couple of witnesses for just the cost of the fees (c.£250 last time I checked) and be just as much married as someone who's had the full-on castle and 3 course meal for 100 guests shebang, but I am guessing you want at least some kind of party too?
There are definite ways to cut costs significantly but will usually entail some degree of compromise that may or may not be acceptable to you. The biggest one is usually to significantly cut down the numbers of guests as venue hire, food and drink are usually the biggest costs of a wedding reception. If for instance you can reduce it to c.20 that opens up your options to either host something at home or a relative's house or to go to a local restaurant. That way you can probably have a pleasant 2 course lunch and offer everyone a glass of wine or soft drink to toast with for around £500. Or for the same amount a lovely buffet, afternoon tea or barbeque at home (buy catering style platters from cost co or even get a small local caterer to do something for you, or order in a big takeaway curry/pizzas/fish and chips, and hire crockery and glasses so as not to have to wash up afterwards).
Obviously if you have big families on either side or would want lots of friends there then this ramps up your guest list significantly and so your costs, but it is still possible to throw a party reasonably cheaply if you abandon wedding 'norms'. Village halls or other non traditional venues are a good shout, basically anywhere that will allow you to self-cater and bring your own booze (a cash and carry run with a van is often a good idea for the latter). Again move away from the 'traditional' (which has only really become the norm in the last 30 years or so) seated 3 course wedding breakfast and think of things like hog roast, fish and chips, curry, afternoon tea to easily cater for bigger numbers on the cheap. Or pubs will quite often do you a good deal for a private room and buffet with a certain amount put behind the bar (although capacity usually slightly more limited), particularly if you are able to do it midweek (or in London square mile pubs are often deserted at the weekend and will do you a good deal for a wedding party although probably not 'cheap' in the normal sense of the word). Or if you have a friend or relative with a large garden or land a marquee can be a reasonably cheap way of accommodating large numbers although the headline cost of just hiring the marquee isn't everything, outside of the height of summer you'll want heaters and lights which means a generator/electric hook up, chair/seating hire, crockery and glasses hire, and you'll probably need toilets, sound hook-up if you want to play music or have speeches etc.
Everything other than the food/drink at a wedding reception is optional IMO so you can cut costs according to your own ideas as to what's worth it and what isn't. Often people advise that you look at what talents your friends/family have and ask them to do things for you instead of gifts for the wedding (e.g. can someone make the cake, do the photography, drive you to/from ceremony in their nice car, make the dresses, do the flowers, perform at the reception or act as DJ) which can be great but obviously only works if they are willing and able, I've seen it go horribly wrong where people feel pressured to say yes but then lack the budget, skills or equipment to deliver the 'professional' standard of service the couple were wanting which can lead to ill feeling all round, so be careful there and obviously adjust your expectations accordingly, your mate Pete who tinkers with cameras for his landscape photography will probably take nice snaps but maybe isn't used to organizing group shots and won't have time to edit 100000 photos into a perfectly presented and filtered album the way a professional would do it, your Mum may make a lovely lemon drizzle but a three-tiered beautifully decorated concoction to serve 150 may be more than she can really manage at home etc.
Some other ideas would be:
-Second hand or hired wedding clothing, there are specialist shops which sell or hire wedding dresses, and suits can of course be hired for the men. Bridesmaids dresses can be hired too, or bridesmaids/children's outfits can be bought second hand off vinted. Or some people I know have had success buying cheaply from China but this seems risky to me. Or abandon traditional wedding clothing altogether (we did this) and wear something you already own or buy outfits you already own. Bridesmaids, best men, page boys etc are not compulsory or if you have them can wear clothes they already own.
-Flowers and other decorations are nice to have but not essential, silk flowers are often cheaper than fresh and can look nice (plus don't die if you want to keep as a keepsake).
-Favours are totally unnecessary and often go to waste. If you want to have them consider making them homemade and/or consumables.
-Entertainment - live bands/singers and a DJ are nice to haves but can be expensive, a playlist streamed through speakers is fine especially at a smaller/more informal party.
If you post your rough budget, area of the country and wishlist (no of guests etc) people might have some good ideas for you?