I have vague memories of this series - from what I can recall it was a very "grown-up" sitcom and had a slow and sad sounding song as its theme. I recall that the opening titles featured faceless wooden puppets without strings - an obvious play on the title.
No Strings must have been a flop if a second series wasn't commissioned. I know there were a few late 80s sitcoms (many on BBC 1) that didn't make it to a second series - the David Essex comedy The River being one.