+1 Palerider.
Talk directly to a bank and get an indication of whether they will approve or not. If its a straight forward case you'll have enough to show the vendor you'll be able to close very quickly.
However, from your previous post last year, your financial history might be a bit complex which might be taking the time?
Make sure your solicitor included a 'subject to mortgage' clause in contract - if not you can't simply walk away.
If I understand your post, you might borrow money from a relative, effectively buying the house with cash, and you are wondering if it's then possible to get a mortgage later?
It might be possible, but it depends on your circumstances. You would need an equity release mortgage which the banks haven't pushed since the crash, but they are available. However, you'd need to pass their underwriting and one standard criteria would be proving mortgage repayments over a period, which you won't have.
I'd pursue the straight forward mortgage route first, rather than getting more family loans and not being sure if you can refinance them later.