I got Windows XP professional service pack 3, and when I start the game, it does not begin; there's no program running for it, and the play button in Galaxy is greyed out as well.

Exact same situation here, using an old machine with XP, downloaded & installed through browser initially, game wouldn't launch. Then got XP service pack 3, got GoG Galaxy, downloaded & installed again, game still won't launch! So frustrating.

I tried to run it on 3 xp machines: my desktop with ati card, my wife's laptop on intel extreme and my father's with nvidia mobile. The game doesn't start on any of them.

Problem may of course be that I was the one who installed all software on those pcs and one of the soft I usually install - conflicts with gog release. Hopefully gog will update their installer and the game will eventually run

What happens when the game doesn't run? I'm having issues on a Windows XP VM at the moment (I run Linux, so the XP VM is my fall-back). The behaviour I'm getting is Explorer maxing out my CPU as soon as I run the shortcut.

Tried the game at 2 different PCs:
Game doesn't work with Windows XP but work flawlessly with Windows 7 PC and Intel HD Graphics.

It seems to be very sensitive to what other software you have running in the background. Shutting down anything that you don't need might help.