I think your initial assumption, that it was the opening theme until it goes to Bright Eyes, was correct. The theme song changes in season 3, and Andrew Lloyd Webber's credit is also dropped.
Perhaps the Mike Batt credit is simply a reference to Bright Eyes at the end of the intro, or maybe there is an alternate version of the song that wasn't publicly released.
If you're halfway through season 2, you won't encounter any further lyrical songs throughout the rest of the series (apart from the usual Bright Eyes during the end credits).