When do employers ask for references?

I was offered a job about 3 weeks ago with a conditional letter, the two conditions being receipt of 2 references from current/previous employer, and proof of right to work. I accepted the offer and have signed the employment contract, and when I asked what information they needed from me before I start, the hiring manager said just a copy of my passport.

The reason I ask is because I have been working as a freelancer for the past 3 years, and was at university for 4 years before that. So technically my most recent employer was a company I worked at 7/8 years ago. I make it clear on my CV that I work on short term projects through websites where sometimes I don’t know the client’s personal details etc, so while I could get references from these projects I don’t think they would be worth much. Similarly I don’t know if the company will be satisfied with academic references or work references from so long ago. I’m pretty confident that it would be fine if they do ask but there is a small part of me thinking that there’s a 1% chance that the offer falls through because of this, and it’s concerning because I am moving cities for the job and signing an apartment lease etc.

I have checked job websites and their listing is no longer on there, nor is the job being advertised on their website anymore. I assume that means they are committing to me, but if so then I am wondering why they would wait so long to ask for references, even after the conditional letter? I have spoken to my direct line manager who is under the impression I’m starting in just over a week, so I’m hoping everything will go smoothly but I would just like to be 100% sure.

Is this normally how the referencing process goes? Surely from their side of things it makes sense to check references straight after the offer letter, just in case they need to line up alternative candidates. Any advice is appreciated.