I went to my GP. The first thing she was concerned about was the blood loss. She offered to prescribe me something to stop the bleeding, but I was concerned (and she thought it was reasonable) that just making the symptoms less apparent wouldn't be helpful when they might be necessary to make a diagnosis.
The first test she did was take a urine test (for pregnancy) and blood tests (for blood count, thyroid function and glucose levels). These showed nothing about the cause, but they did show I was anaemic from blood loss, so she prescribed me ferrous fumarate for that.
The next thing she did was a speculum exam and cervical swabs (the sensation of which was peculiar but not painful). She said straightaway after the speculum exam that I had some cervical erosion (aka ectropion), but that she didn't think that would be the cause of such prolonged bleeding. The swabs came back showing that I had an extensive infection of thrush. She said that given that I have the cervical erosion, the thush infection on that endometrial tissue could be causing the bleeding.
I thought this was weird because I've had thrush before and I thought I would know when I had it because the itching drives me completely up the wall. I haven't had either any itching or any of the characteristic discharge. But in any case, my GP's plan is to treat the thrush, then if that doesn't help, the next tests will be ultrasounds to look for structural abnormalities.
Everyone I've been in contact with during this set of dealings with the NHS (the national health service here in the UK) has been magnificiently respectful, knowledgeable, compassionate and has genuinely wanted to help me. They appreciate that it's a problem for me to bleed so heavily even though I'm not experiencing any discomfort and am not in any real danger. From receptionists to nurses to doctors, they have all been deeply courteous and gone far above and beyond in order to help me and make me feel comfortable.
tl;dr - Apparently thrush/YIs can be pretty extensive without causing itching and discharge. They may also cause prolonged and heavy bleeding when combined with cervical erosion/ectropion. And the NHS is great.