This app was mentioned in 3 comments, with an average of 1.00 upvote
Link to the app on the Google Play store
A small app which lets you manually record the times you sleep/wake and provides many graphs which can show useful information. I use it to get an idea of what my sleep deficit is and to try to predict my sleep patterns for the next few days. This is a screenshot of the graph I find most useful: https://i.imgur.com/nynIWfZ.png?1
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I think https://www.sleepgraph.com/ will do what you need. The import of the data is done manually by them, but they're very good and happy to help.
I use an Android app called Sleepmeter Free to record and display my data, and they were able to take the export from that app and import it into the website.
No probs :-)
It sounds like you're at the point I was in around 2008. I got lucky when my old GP retired and I accidentally came across info about N24 from a colleague at the same time. The new GP was much more willing to listen to me (though we still had to rule out the common causes). I'd been on and off anti-depressants for over 10 years at that point, and had been having sleep issues for over 5 years before that.
Something to note is that there's every chance you do have depression, but probably it's a secondary from the messed up sleep patterns (seratonin regulation etc.) and these will compound the underlying problems with sleeping and waking. When I was still trying to "be normal" when I did get to sleep, it'd almost always be for a minimum of 12-hours. Sometimes it would be 2 days. Even now that I'm "free running" (sleeping/waking to my internal clock) it still took a few years for my sleep duration to fall to a more normal level of about 7 hours, and I still get the odd random 36-hour sleep for no apparent reason.
I'm getting off topic though :-)
If you can't convince your GP to listen to you, it's worth asking to see another GP - you have the right to and you can even see one at another surgery. Don't be afraid of moving to a new GP in a different surgery if that's what it takes - if your current GP is taking a holier-than-thou attitude anyway (which some do) because of either a lack of information, out of date info or a superiority complex then they are not putting your best interests first.
Do remember too that although your GP may act like it, they are not at the top of the food chain - the surgery will have a practice manager who you can speak with to air your concerns, and they will have a manager, and so on. If you feel that you're being ignored, you can take matters to a higher authority.
You might encounter another stumbling block - years ago, when circadian disorders had not long been recognised as a "thing" they were classified as psychiatric. Of the few medical professionals in the NHS who are aware of them, some haven't gotten the info that they're now classified as neurological.
Arm yourself with information and don't give up. It will be exhausting at times trying to convince them, but getting a diagnosis is worth it even if for nothing more than to have that piece of paper which says "I'm not making it up". It also ensures that another circadian rhythm disorder diagnosis is recorded and gets it noticed that fraction more by the NHS.
The CSDN website is packed with helpful info, and there are support groups too - there's the Niteowl mailing list and there are Facebook groups too, all linked to from the CSDN website. We're scattered around the world of course, but I know of 3 others including you who are definitely in the UK. I'm sure there are more but I mostly hang out on the N24 group and it seems there's fewer of us than there are DSPD.
Feel free to message me if you want to, and don't forget the support groups (including this sub) - just knowing that there are people who understand makes a huge difference sometimes :-)
Edited to add: You might find it useful to log your sleep with an app. I don't like using the ones you stick under pillow so I use a manual one, Sleepmeter Free where you press a button when you're falling asleep, and press it again when you wake up.
You will almost certainly still need to do paper charts though for a sleep lab, as they'll want to put them in their notes. A printout of an app logged chart might be good enough though these days.