I can't say specifically about Barnie, but Australorp crosses TEND to have Australorp feathering, with a few coloured feathers to reference the other parent, and may or may not have the type of the other bird.
I can say for certain that Araucana/Barnevelder (="Barnacana") rooster over Australorp hen produces a black-feathered bird with Barnacana-ish type (stocky and low-set but with some bizarre curves in odd spots from the Australorp), muffs and peacomb, and green eggs. Black legs and beak, red face and small comb. Can't see lobes though the muffs
Chicks were, if I recall correctly, Australorpy penguins but with the Araucana muffs. Very, very cute little fluffy handfuls.
Your mix would _probably_ be a stocky, medium-sized black bird, maybe with some brown or maybe even laced feathers around the hackle or chin, and _probably_ lay light brown eggs of reasonable size. Leg colour is dicier and the genetics people would have to get involved on that - could be anything from black through to the Barnie yellow, including a sort of dusky brown/grey. The black Australorp eye colour tends not to carry through in crosses - it's the first giveaway in telling an Australorp cross from a pure - so it would be more like the Barnevelder's - orange/yellow, as I recall.
The cockerels would be brilliant eating as both breeds are excellent eating birds.
The progeny are unlikely to throw true to either type but it can happen. _Their_ offspring, however, could throw either way or to their parents mixture; it would depend on what you put the first-cross birds back to.