Finally I solved the issue myself.
After repairing everting I could think off, like installing an older graphics driver, re-installing Steam, Steam VR, F1 22 and F1 23, disk and Windows checkups etc the solution was rather simple. Too bad the EA service Desk did not advise this...
Go to Documents\My Games folder. Rename the F1 23 (and, in my case, the F1 22) folder to F1 23-old and restart the PC.
It worked again! Mostly that is.
Sometime it also helps to change the VR settings of the game. It probably overwrites some corrupt files in that case.
Renaming the F1 23 folder might help for other issues too