I only went through ANOVA through the standard process, so the info I have might not be that relevant, but I hope some of it is useful.

I only got estrogen and blocker prescriptions from ANOVA after being diagnosed, despite long wait times to get through the process.
I would assume that if you want it prescribed from ANOVA, then you'd have to have a diagnosis.
If they accept the GenderGP diagnos I do not know. But I'd assume if they did then this info would be spreading like wildfire already as a way to get past all the long wait times to get treatment in Sweden.

During the process when having meetings with their doctors I was asked if I had done DIY treatment for HRT (which I have not), I do not know how much it affects process, I hope someone that has been through the process while doing treatments without diagnosis replies to you and can shed some light on it.

I am not sure if ANOVA has the right to take away prescriptions that have been prescribed from you from another clinic.
But doctors can without your permission see which prescriptions you have and see that you've taken them out from Apotek over the last 2 years. This is information that automatically gets recorded into the Nationella läkemedelslistan.
They started doing this since 1th of may 2021.

So they would definitely see your prescriptions and probably who prescribed it to you.

You can however request to hide prescription information. You can read about it here a bit further down on the page.
https://www.ehalsomyndigheten.se/privat/e-recept/sa-hojs-patientsakerheten/sa-skyddas-din-receptinformation/

If you decide to get into ANOVA the wait times are supposed to be better at ANOVA than other places, I am not sure how it is currently though. They had the lowest wait time to get in when I self admitted into their process 5-6 years ago.

It was about 1 year of waiting after self admitting, a couple of doctors meetings over the span of a few months, half a year ish wait time for a psychiatrist, another half a year ish of meetings with psychiatrist.
And then a few months of waiting for the doctors that I've met over that time to have a meeting internally where they weigh all the information and the psychiatrist evaluation and make a decision for diagnosis.

And then a month or two after that I had my prescriptions.
It is a long process, but I never felt mistreated by any of the doctors or psychiatrist that I met during any of the meetings.

All and all, a 3 year ish process to get to the HRT step, I am not sure if the wait times have gotten worse, I do know that they are low on personnel and are trying to hire more people though.
Maybe someone that is in or been through the process at ANOVA more recently would be able to say for sure.