You can always stop and ask, but if your bag isn't checked through,
you need to pick it up, which in this case means entering the UK at
this point (and I'm assuming flying domestically to complete your
travel?)

You cannot trust airline staff to know the answer to this.

In traveling EU to US via Canada, I was told that I must collect my
check-on luggage in the *first* US stop, and re-check it. (this
sounded reasonable because i've had to do that in the past). But it
was bad info. *Two* staff members in two different cities along the
way gave these same wrong instructions.

I went to collect my bag, but it never appeared. I was forced to run,
stand in line, and re-enter the secured area to catch the next flight.
The bag was ultimately checked through the first city traversed in the
destination country, and arrived at my destination. I was needlessly
advised to go through TSA, and risk missing a connection.

Travelers need a more certain way of knowing, without relying on
airline staff. Sometimes there is a customs "pre-screening" check,
and sometimes not. Sometimes there are multiple international
airports (read: airports with customs). In some cases there is a
customs service for the destination country doing a full customs
process on the soil of a foreign country.

How can we know? There must be an algorithm.