Yes, responsibility normally transfers to the new owner. When you buy a property, you inherit any ongoing planning breaches unless the seller resolved them before completion. That includes things like unauthorised UPVC windows in a conservation area. The council does not need to “restart” the process just because ownership has changed.
However, enforcement works differently depending on the rules in your area. If there is an Article 4 Direction covering window changes, the council can insist on timber replacements and can take formal enforcement action. If there is no Article 4 Direction, they often rely on conservation area status alone, which gives them some power but it is weaker. They can still serve an enforcement notice for unauthorised works to the exterior of a building in a conservation area, but whether they actually do so varies a lot by council and by how strict your conservation officer is.
The bigger issue is that councils often threaten action verbally but never put anything in writing or follow through, which sounds like what you are experiencing. They should put any enforcement steps in writing, and until they issue a formal notice, you are not legally required to do anything.
In short: yes the responsibility sits with you now, but unless the council serves a proper enforcement notice, it remains just a warning. Checking whether your area has an Article 4 in force will tell you how likely they are to push it. If not, they tend to advise rather than demand.