'Fraid not, karentollet; the first episode of 'Only Fools And Horses' was aired back in 1981 with a sort of brassy jazz theme tune, but without any vocals whatsoever. Definitely not The Hollies! Thankfully, from episode 2 it opened with the familiar theme tune we know today. Sorry about that - 'He Ain't Heavy, He's My Brother' would have been much more appropriate, if you think about it ...
As a footnote, I always thought the theme tune was sung by Rodney, whereas it turns out that it was actually voiced by John Sullivan, the show's creator. What a plonker I really am!