Awkward conversation with new neighbors about replacing garden boundary walls..

They don’t have a concern about it falling down ..

can the OP post photos ?

Sure they can. As it stands, you are the one with a problem not them. Your surveyor's report is an exercise in arse covering on his part. It describes the walls as both rectified and recommends replacement. It is a report written to facilitate your purchase, not the actual state of repair of the walls.

You mention your 2 daughters and the risk incipient in their playing in the garden and a potential collapse. Again, that's your risk, not your neighbours.

Also as others have alluded to, the fact that there are walls in place between properties doesn't actually mean they are shared/party walls. Shared walls are built upon the boundary line, the map and the wall must match. Now in an estate, the likelihood is that the walls were built by the developer and are shared walls.

That however doesn't preclude the possibility that the wall as built, is entirely inside your boundary and as such, is entirely your problem. It's unlikely but, possible that the previous owner built the wall to enclose the property rather than as a shared boundary.

The advice offered by others regarding the actual condition of the walls, how they hold up to a winter and the timing of your request to your neighbours for co-funding a rebuild are all good. Noone wants to spend money before Xmas, and even fewer want to spend money where they don't perceive either a problem or a benefit.

I rebuilt a boundary wall between one of my neighbours and I as part of a shed build a few years ago. The existing wall was cracked but stable. Its alignment included an angle that kicked out the wall in a way that meant building along the line was awkward.

I spoke to my neighbour, asked him if he had any objection to me taking down the existing wall and rebuilding it on a straighter line inside my boundary. He agreed, signed an agreement to the effect (he gained 1.5²sq Mtrs of garden. Then I built the wall and bore the cost. My problem, my solution and my cost. If the wall that was there had collapsed? If I chose to rebuild it as it was? Then yes, asking for a contribution would certainly have been fair. Insofar as it would have been towards restoring a collapsed, shared wall to it's pre-collapse state.

Deciding to preempt that potential collapse and demolish & rebuild before any collapse occured though? As sensible as that seems from a precautionary principle standpoint, if the neighbours are at no imminent risk other than "collapsed wall"? I can certainly see why they'd prefer to wait until a collapse occurs and then repair that portion, than cofund a demolish and rebuild.

Have you any pictures of the wall and cracks, maybe you can build up some supporting block piers or if there are extisting piers run some bracing between them. It might save you rebuilding the wall.