First thought: your friends should make certain that the colleague they are intending to move in with has the right to sub-let part of the accommodation to them. The whole arrangement will need to be open to INZ to see, for their partnership evidence, and it would not help if, later on, the head-tenant didn't want to share details with INZ.
Second thought: they will need to have a letter from the head-tenant (and possibly, anyone else also part of the house-share) that they have private shared space as a couple, showing that they are not just individual house-mates.
It shouldn't cause a problem, not having their names on the lease or utility bills, as long as there is a formal trail of evidence to show that they are routinely paying their share towards the expenses of the accommodation. This could be the written agreement you mention, coupled with tagged mentions in their financial records of what the various payments are for.