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According to this comment in the MECoOp Reddit thread "Ammo power interference", there are two types of combo primers, "biotic" and "tech". Biotic and tech primers can both exist simultaneously on a target, but they can only be detonated one at a time by two different detonators, and the tech combo apparently always detonates before the biotic one:
Ammo powers that prime are considered tech primers. Tech primers layer
over biotic primers, so a detonator hitting a target affected by both
will detonate the appropriate tech combo. If the target is then
detonated again, a biotic explosion will occur.
So ammo powers won't override biotic primers, nor will they prevent biotic primers, but in order to detonate both primers, the target must be detonated twice, and the tech combo will always trigger first.
However, I'm not sure if ammo powers will override other tech primers, such as incendiary ammo overriding a rank 6 Inferno Grenade on a target. The same commentator said that
A target can only be primed with one biotic and one tech primer at a
time (most recent takes precedence if a second primer of either type
is applied to a target already affected by a primer of the same type).
According to that, it seems like a rank 6 primer such as Inferno Grenades could override an existing incendiary ammo effect on a target (it's not "blocked" by the ammo power), but it's not clear to me whether it's also true the other way around.