Healthcare: Correct english term for an urgent referral

In the UK the term used is urgent referral.

The following text is taken from a local department of the National Health Service (the government body responsible for healthcare treatment in the United Kingdom). It concerns advice given to General Practitioners regarding how they should make referrals:

Referral Guidelines

Urgent suspected cancer (USC)

All urgent suspected cancer (USC) can be emailed to [email protected]. -- Croydon Health Services, NHS.

The Cancer Research UK organisation also makes reference to this specific type of referral in their advice for patients:

What is an urgent referral?

Your GP has arranged for you to see a hospital doctor (specialist) urgently. This is to investigate your symptoms further. You might have some tests to find out what is wrong and if it could be cancer.

Your appointment is arranged with you by phone or post. Or you might book your own appointment online. Your GP will explain what to do if this is an option for you. - cancerresearchuk.org

Regarding the point you make in your question that:

...there is a big difference in the level of urgency between "urgent" and "emergency", with an emergency being way more urgent than something which is just "urgent", at least to my ears.

I agree, but I don't think the wording is accidental on the part of the healthcare professionals. My thinking is that they do not want to create fear or panic in the patient, and an emergency referral would likely do that more often than an urgent referral would.

Edit:

Also a referral implies a passage of time between the referral and being seen by a medical professional - as @MaxWilliams astutely points out in his comment, this doesn't really suit the use of the word emergency:

..."Emergency" is reserved for cases where someone needs immediate medical attention - they've had a heart attack/stroke, or suffered a severe injury, for example. So, an emergency is ambulances with wailing sirens etc. You would never have a referral in this context: "You seem to be bleeding heavily from your neck, I'm referring you to a neck specialist".