This is a very simple method of trying to solve the problem, must point out that I would do it this way or use the same products, but I have the benefit of the products and equipment to be able to do it differently. So I have to advise the DIY way
Nathan, It looks like you've been removing carpet that has covered the tiles and probably prevented them from airing and drying out sufficiently.
Initially you'd be wise just to get all the old floor covering up and out the way. If there's any thick powdery white powder, vac that up and follow up with a stiff scrubbing brush. This is just to remove the old dry efflorescence that is laying on the
tile, vac what comes up.
If this tactic is working but leaving some left behind then you may need to repeat.
Then I would leave the floor for a few days to air out and see if the efflorescence reappears. If the efflorescence does reappear then further investigation would be required as either there's still damp in the tiles or the mortar bed, so you'd either have to remedy the source of the damp and leave it to dry (preferred option) or apply products that inhibit the efflorescence; I've tested loads and never been impressed by any of them.
If no efflorescence reappears then again a scrubbing brush and lithofin KF victorian
tile cleaner will remove a lot of the dirt, you'd be wise not to let the product dwell for too long, no more than a couple of minutes or you could reactivate the efflorescence. Work in small areas and use more than one pass.