Unless you’re willing to wait out the landlord taking care of this, and live with roaches, I’d be doing some measures to improve the situation myself. Every day you leave the roaches undisturbed, the infestation increases- roaches propagate at crazy rates. Roaches themselves carry disease, and so do their feces. UGH.
If the neighbors are nasty there’s nothing you can do about how they live, but you CAN go to a home pest control store and buy whatever they recommend to spray/bait your place, (with the cats in mind). Because I hate to tell you, but even if they spray, nasty neighbors will have a continual problem. You’ll have to do battle on your own turf.
Start in the kitchen. Roaches love to hide under the fridge and stove- pull them out (be prepared to shriek, because you might find a small army of the little buggers) clean up whatever grime you find, and put out the roach bait. Agree with what AK said- you have to be meticulous with never leaving anything out.
