If the flights are set in stone I’d strongly encourage the op to find a way to fly with hand luggage only even if it means sending a suitcase separately and at a cost.

with no checked bag this will just be a case of walking to the next gate by gate closing time, so safe bar huge delays. With checked bags, we are talking clear immigration, wait for bags, clear customs, recheck by luggage drop closing time, clear emigration and security. It is far more risk points lined up. Oh and through checking won’t happen (in that direction).

delays happen (what if there is bad weather or a strike in the uk?), flight times can change (3hrs can become 2) and the flight the op could lose entirely is the expensive long haul.

there is probably a 90% chance the op’s plans work ok, but the remaining 10% is unmitigated disaster rather than mere inconvenience.