literally all an estate agent will do is hook the seller up with someone who wants to buy and then gets paid a big fat wodge of money for it.
Therefore if you already have someone who wants to buy, you can dispense with them completely, if your nan is happy with the price the neighbours want to pay.
What might be an idea is to get an estate agent to value the place for you which is a free service and see if it is what the potential buyers want to pay. you dont have to then put it on with the estate agents, and potentially you could reduce the price by a couple of thousand as this is what you would ordinarily have paid in fees if you sold through the estate agent.
Once you have negotiated the price with the neighbour, you proceed how you normally would do: get a solicitor. There is no more chance of the neighbours pulling out than if you used an estate agent, or if you had another buyer (in fact i would say there is less chance if the neighbours are buying it.