Couple things to try :
1. press Install on the launcher menu (mine did not change to play the first time either but pressing Install again updated it and after that it now says play (I installed to a different folder so think the launcher could not find the right folder the first time but clicking install made it update.
2.) go into the folder that you installed the game to and launch the game using the TESV.exe instead of Skyrimlauncher.exe and see if it launches that way.
Thanks for the reply, much appreciated!
1. I've clicked the install button, which pops up a steam dialog box saying it's preparing to run Skyrim, and then drops back to the skyrim splash/start window. No luck there...
2. Running the exe directly DOES work! I had to fiddle with the ini file to maximise it and tweak the graphics to something that is playable on my older rig, but it does play.
I can play the game now, but not through Steam. It's still not a happy solution but at least I can do something until the main issue gets sorted somehow. Thanks for the tip there!
I still want to get it working fully, though.