I believe I've found a solution to this issue.

Go to: C:\Program Files (x86)\Recursion\RecursionTracker

And open the file user.config in a text editor (eg. Notepad, Notepad++, SublimeText)

Search for 'AutoLogin'. For me the value was 'False' even after enabling it in settings. I switched it to 'True' and Recursion now seems to autologin when it starts on Windows startup.

I haven't tested whether this will persist through changing the configuration inside Recursion yet.