The "correct" order varies with your situation. One of the things affecting your situation is which country you are living in. Don't try to apply a simplistic "one size fits all" even within a country.
Looking at the differences between the US and UK, you've already identified one - student debt isn't typically as large. That doesn't affect the priority of it, just means it is easier to pay off. (By the way, it isn't always best to pay off student debt - sometimes it is low interest and you are better off keeping it.)
Another difference is that US emergency funds are mostly there to cover a loss of job. In the UK people are generally less likely to be laid off, and if they are there is more likely to be a severance payment and unemployment payments are likely to be higher. In the UK there is no concern that you will be unable to cover medical bills if you are laid off, as there is in the US.
All of this means your emergency fund might not need to be as big. But remember this varies with many things, not just country. If you are in a highly desirable profession, or one with very stable employment, then you don't need as big an emergency fund whatever country you are in.
Also different tax regimes mean different priorities. In Canada many people use their retirement savings as an emergency fund to some extent, becuase retirement savings can be withdrawn relatively cheaply in an emergency. They can also be used to fund a house purchase, so they get higher priority. My knowledge of British pensions is out of date enough that I won't comment.
In Britain it is harder to save for a mortgage than the US, and many people prioritize this over retirement savings.
But my main point is this - don't take someone else's generic bullet list and follow it - work out what you need in your own situation.