If an arriving player gets there, AFTER the tackle is made (Knee of the floor is tackle made) then your "immediately" is up. If he is grabbing the ball before the tackle is made then he must release, an go back in. Of course all this depend upon clear and obvious.
I'm not talking about an instant later. YOU referred to your knee hitting the floor and that is tackle made. Judgement call. But there it is.
Its quite likely that Blackberry is convinced he is right, but I disagree.
IMO, the tackler's
"immediate" has a shorter life span than that of the tackled player.
Law 15 says
The tackler must release the tackled player and ball immediately
The tackle assist must release the tackled player and ball immediately
The tackled player must release the ball immediately by placing passing pushing or releasing
It does not say what order this must happen, convention seems to be that the tackler's and Tackle Assist's "immediately" expires first. If the quickly arriving player is trying to take the ball from the tackled player, while the tackler is still holding on to the tackled player, I'm pinging the tackler for not releasing!
This can raise another problem however.; if we expect tackler to release the instant the tackled player's knee touches the ground, then we need to start being vigilant on tackled players getting back to their feet again. Perhaps the emphasis need to change from assuming the ball carrier hasn't been tackled to assuming that he has, and penalising accordingly. Its not much to change and its easy to sell.
Player: "But ref, I wasn't held"
Referee: "That's because the tackler released you as he is supposed to"