Having looked at the issues in greater detail, and in consideration of the expanded facts, I rewrite my advice.
The caveat I mentioned does not apply to unregistered couples - only to those who have entered into marriage or civil partnership. My mistake! Unregistered couples (logically, of course), cannot acquire matrimonial or partnership property if they don't enter into either of those legal arrangements.
So your friend and her ex girlfriend are given a status only slightly above that of former flatmates, and have substantially inferior property rights to those who are divorcing or dissolving a civil partnership. That's the good news for your friend. She is the sole owner of the property and her former girlfriend has no right to it.
The downside is that since part of the price was paid for by the ex girlfriend, she is likely to be entitled to a share of it's value. How much? I would guess that whatever the share of the original investment was to the share of the property's value at the point of separation. So if A and B buy a flat and A pays 90% of the price and B the remaining 10%, then B is entitled to 10% of the property's valuation as it was on the date of separation. As happyjo has pointed out, a constructive trust would appear to exist or, in Scotland, grounds to sue under the law of unjustified enrichment.
But I would argue that this supposed right to 10% of the property's value would be offset against B not contributing anything else, living rent free, etc.
So should you agree to mediation? I can't really see how it would hurt and anecdotally I've heard that Judges frown on those who go to court without trying to resolve disputes.
But....