Weird question - can you go through English Immigration without leaving the country?

I don't think it's that simple. This is a serious violation. Once it is recognized (by asking for the Visa stamp with a document with a much earlier effective date) they will be asked about what they have been doing in the interim. It would almost be better to take the Eurostar and come back for another 90 (?) days. But they are still at risk, if they are asked about their legitimacy (jaywalking? rock concert security? Lost property inquiry? getting mugged? ... ) before they finish the educational program.

I'm saying that your question does not solve all their problems. What did they hear when they went to the Dean of Students (or the Limey equivalent) and asked for advice? When I was a freshman, I put off going to the Dean too long when I had a check forged against me. As a Professor once said to the class, in a different context: "Deans are mechanics. They fix machines."

Edit: What I meant by my second paragraph is that you can’t be the first student to have done this. The Dean’s office probably knows people in the Immigration office who can fix it for you, because they import students every semester. What’s the worst that could happen with the Dean? That he tells you you were naive and careless? You were!

If you can see the connection: When you go to a show at a regional theater these days, the program credits include the theater’s list of 3-5 law firms. The third one is “Immigration Counsel.” (Who sez, “The unions are killing Show Business?)

I think rather than trying to fool the system and risk being "caught" or denied reentry (they are under no obligation to allow you back in if you leave), I would admit your error and go to what I assume would be the Home Office, specifically the office of Immigration and Naturalization in London and sort it out.

I do not think the student did anything "illegal" since they are legally there on a tourist allowance and apparently already have the Student Visa, just did not do entry properly for the visa

They could always contact the US Embassy for guidance (assuming US Citizen) but you need to find the Immigration office.

Contact details for general immigration queries

https://www.gov.uk/contact-ukvi-inside-outside-uk/y/inside-the-uk/general-immigration-enquiries

There is apparently no specific procedure in place to regularise people who have entered the UK via the egates as visitors that should have got a passport stamp for their Short Term Study Visas, unless this is done immediately before leaving the immigration hall by a Border Force officer. The only solution on offer presently is to depart the CTA and re-enter properly before the study is due to commence.

This is not a new issue in a way as people have been caught out for years by entering the CTA via Ireland, where again this bypassed the validation stamp stage.

Thanks everyone, I wasn't sure anyone would respond so this is helpful. I'll pass it along.
=Tod

but they are still at risk, if they are asked about their legitimacy (jaywalking? rock concert security? Lost property inquiry? getting mugged? ... )

No such thing as jaywalking in the UK, you can cross the road, walk alongside the road or even in the centre of the road whenver you feel like it. Obviously common sense applies however the law states that pedestrians in the road have overall priority (except motorways and roads that don't allow pedestrian access).

No such thing as jaywalking in the UK ...

Apart from Northern Ireland. Although enforcement is rare and mainly confined to cases with consequences.

In Northern Ireland, you are not allowed to be middle of the road. It's either one side or the other.