I am including this here to clear up a few misconceptions.

ABBA never appeared on The Basil Brush Show in person.

The misconception probably comes from the fact that it was so long ago and we were all just little kids ourselves with memories wanting to believe they were on the show - and none of us had video recorders back then to prove it one way or the other. Another thing to fuel this misconception is that they were pictured with the Foxy One during their promotional trip (for Arrival) to London in November 1976 - around the time when they *could* have appeared on the show.

The are listed in TV schedules and TV archives as appearing on The Basil Brush Show in 1976, but it was the performance of Money, Money, Money from ABBA-dabba-dooo!!

The reason I know this is because The National Film Theatre in London held an ABBA at the BBC evening on 20 December 1995 and The Basil Brush Show was on the list of appearances to be shown. I was beside myself in anticipation that nearly 20 years later I'd get to see these clips again but it turned out to be nothing unusual or unique.

Update: 17 August 2023
ABBA was the first performance on the show. They were introduced by presenter Roy North (Mr Roy) and Basil Brush himself.  It was indeed was the performance of Money, Money, Money from ABBA-dabba-dooo!!  This information was from the whole programme on YouTube which has since been removed. The clip was a film copy of something made on video, and you can see the difference in quality, cropping and distortion when you put the same frame side-by-side.

The Basil Brush Show; 1976
The Basil Brush Show; 1977

Thanks to Carl Magnus Palm and Stuart Rowlands.
YouTube link c/o Retro British Television