I have not heard of the name neighbourhood theatres used in this country,the term second run comes to mind from days gone by.As to the actual length of films altering after there first run and going to the smaller cinemas,all the ones I showed were as per their first run.
There was of course the films that did not get a general certificates per Don's reply and were licenced by the local authority or watch committee as it was I think,they determined the certificate and any cuts to be made,in Plymouth if I remember Ken Russell's The Devils was certified locally but as to cuts in that film not sure.
For long films,roadshow print type ,with overture play in ,intermission etc they were shown in their entirety as the director intended to the best of my knowledge.
Maybe others know different,there are stories of reels being missed out etc but that was a long time ago,I'm sure some on here could tell a story or two of their projection days when things did not go as planned and perhaps film length did get altered more by accident than design.
Obviously prints were altered for use on TV etc and if you look at the IMBD website on the technical info of a film there is usually listed all the alternate versions for different countries,16mm,check out Gone With The Wind at least six different lengths and four different aspect ratios.