I found the answer by Szabolcs really useful in finding the answer to the same question, so I thought I'd just leave the exact way to do it here in case someone stumbles across it like I did. The speed boost is very good - I'd been relying on this function to get the output I needed and for a list of about 3000 points it was taking about 20 seconds on my machine using DelaunayTriangulation and about 0.1 for the below equivalent!
points=RandomReal[1,{100,2}];
vertexadjacency = DelaunayMesh[points]["VertexVertexConnectivity"]
