Fortunately the OP appears to be very savvy, but lest we should scare someone else in a similar situation who comes across this thread ...

If you reach the UK, with the right set of evidence should the Border Force officer even decide to dig deeper, the chance of being refused entry to the country for the night is tiny.

Theoretical discussions about whether you'd be detained in Border Force's airport accommodation (read: small, uncomfortable room in tne bowels of a terminal) overnight and then either sent onward or back to origin are just that - theoretical. The reality, I expect in excess of 99% of cases, is that the passenger will be tucked up in a comfortable hotel bed shortly after they walk out of the airport.

We can tie ourselves up in all sorts of knots thinking about every possibility. The simplest and most obvious one invariably works perfectly.