Yes, the 806 and the 802 both had the mk 510 dart as did the 700Ds. The 806s were the last batch ordered by BEA and came with more powerful engines (mk 525s?). Then, as you say, along came the Argosy which needed all the help it could get so BEA whipped out the 806 engines and substituted the mk 510 which were obviously more readily available and probably used less fuel. The aircraft were then officially known as type 806x.
It was the 806 which usually operated the services to KWI, BAH and DOH which I guess needed all the power they could get in the summer.