I don't think so, but then I never went to a public school (I assume you mean fee-paying, selective "public schools", not schools for the peons plebs poor people general public). Its certainly not something that some of my friends who went to those schools seemed to pick up. Also, given that a lot of the top universities are likely to have people from all over the world studying there, accent probably isn't as much of an issue as it used to be. But Oxford or Cambridge, I forget which, did catch a lot of flak recently for being too elitist. That may include preferring students from RP-favouring backgrounds.
That said, I do come from the south-west and we all talk like farmers down yur; I also never went to a conventional university. pwhodges or Thrillho might say differently, since they're nearer to one of the big universities (I can't remember which).