It's inevitably a standard "the school has a ghost" legend passed down by students but another potential Cromwell ghost is in the old family home Hinchingbrooke House. The house has supposedly been haunted by a nun and a Civil War type - sometimes a cavalier, sometimes a roundhead depending on teller - and stories are told from sixth former to year 7 to this day about him or her being seen in the house or the presence being felt but as with all these things it's vague and not much is really recent stories or even claims by recent students of things, nor has anybody really found much aside from the occasional medium being fed bogus information Acorah style to test them.
I did once do a totally scientific test though while working in there one evening however. A few of the youth theatre kids were talking about "the ghosts" and how eerie it was in the library, so with me and the six of them there I did the entirely unscientific thing of asking for a sign, a slight breeze seemed to flow through the library and almost turned the pages on one of their scripts so they (ignoring that the front door had been opened in the old place while someone went out to get breaktime snacks) looked at me as if to say "see". I called it a coincidence and then doubled down by saying we needed more to prove you're there.... Right on cue there goes the fire alarm and everyone has to leave the building. While we're outside and I'm getting terrified daggers the fire brigade turn up and point out that some engineers were doing something with wiring in the main school buildings and accidentally triggered it and couldn't get it to reset. Pure coincidences but I'm sure that led to more pupils spreading the legend!