My dc cut his hand at the base of the thumb at 1:45am (as you do!). It's not deep or big but it won't stop bleeding. It bleeds a tiny bit with pressure on, as soon as we take pressure off it starts bleeding again. Not loads but enough that I'm worried it shouldn't still be bleeding so easily/freely after 2 hrs. We've been applying pressure for 15/20 mins at a time.
I called nhs 111 after an hour and am waiting a call back. It doesn't feel a&e worthy but you have to book appts for minor injuries via 111 here so how long do we wait?