- Get a private loan from friend/family
- If possible, borrow the funds from your retirement account (in US, this is a low cost way, not sure how your retirement plans work.)
- Find someone to rent a room in the new house, if you bring the lender a signed lease, they should count a portion of it as income, a few hundred £/mo should cover the 10K you are short.
- Stop spending. If you are truly this close, stop all discretionary spending. No cable TV, no eating out, I mean go "rice and beans" to save that up.
- As OrionDarkwood suggested in comment - Talk seller down by £10K, I don't know if you are looking at a £100K house or £200K, so £10K may not be that high a percent.
- Ask the seller to finance the extra. Sometimes the bank will have a limit, but still allow the seller to finance a second note after the mortgage.
- Ask the realtor (real estate broker) to chip in. In the US, they can take 6%. When a deal is very close, they will often give up 2% to close the deal and move on rather than lose the sale.
- Approach your current boss(es) and discuss a raise.
- Earn extra money. A friend here documented his second job in Deliver Away Debt. He took on a second job delivering pizza on Friday and Saturday nights. (The point is not about pizza, it's about finding clever ways to raise money. £10k might take 1000 hours to earn, maybe less.)
- Combine elements above. A small raise, a few months of second job, etc.
You might also want to talk directly to a bank. If your credit report is clean, they may have some discretion in making the loan.
Note - the 'normal' fully qualified loan has two thresholds, 28% (of monthly income) for housing costs, 36% for all debt servicing. A personal, disclosed loan from a friend/family which is not secured against the house, would count as part of the other debt, as would a credit card. While I don't recommend using a credit card for this purpose, the debt fits in that 28-36 gap.
As Kevin points out below, not all paths are equally advisable. Nor are rules of thumb always true. Not having the OP's full details, income, assets, price of house, etc, this is just a list of things to consider. The use of a 401(k) loan in the US can be a great idea for some, bad mistake for others. This format doesn't make it easy to go into great detail, and I'm sure the 401(k) loan issue has been asked and answered in other questions. With respect to Kevin, if he wrote 'usually', I'd agree, but never say 'never.'