If I had bomb tokens at Amateur Recital, can my opponent use the hold trigger to move their unit off the battlefield before my bombs go off?

No — the bombs fire first, so their unit can't leave before the damage lands.

Holding happens during their Beginning Phase, so as turn player they put Amateur Recital's "when you hold here" trigger on the chain first, and you put the Bomb's Deploy triggers on top of it. LIFO means Deploy resolves first, the Bomb is killed, and its Deathknell ("deal 2 to an enemy unit here") is placed on the chain above the Recital ability — hitting their unit while it's still sitting at that battlefield (Rule 383.3.d.1, 808.1.d.2). Multiple Bombs cascade: the first Deathknell kills your other Bombs there, and each of those deals 2 more.

They do still get the move afterwards, but it doesn't save the unit: Amateur Recital resolves and can move a unit at a battlefield to its base (any unit, not just theirs), while the lethal damage stays marked and the next cleanup kills it at base. Note this isn't something a Reaction-move can dodge either — the Deathknell target is chosen at finalization and "here" is locked to the Bomb's old location, so only an actual damage-prevention or Shield effect stops it. They keep the point they already scored, and if that was their only unit there, they lose control in that same cleanup.