Asbestos is not a problem if it is not breathed in, so you are OK so far. If the tiles are in good shape (solid, stuck to the floor), you can actually cover them up and put a new floor over it (if you know what you are doing).

However, this is the landlord's responsibility. You don't need to be tearing up his floors (what if something goes wrong?) under any circumstances other than he is paying you to do it. And since you are pregnant, I would say not even then. Just not worth the risk. Your new baby is not going to know that their room wasn't decorated perfectly before you went to the delivery room!

The floor is intact for the most part there are a few damaged tiles which is why the floor can’t be covered he said. I don’t need to decorate I’m actually having twins so there’s a chance they could be here earlier than expected. So it’s a matter of getting basic things set up.

Leave the asbestos tile please. Floor right over it; you'll never breath or contaminate a thing and your schedule is less complicated.

The only issue is there are some damaged tiles around the edges. They’re not friable at the moment but they are cracked. My land lord said the flooring company can safely remove the cracked ones and leave the others undisturbed and put wood and quick dry cement where the previous tiles were then put the new flooring over. Does that sound right?

"Let sleeping dogs lie." Yes, it's cliche, but perfect in this application. Sounds like too much unnecessary work to me. Go over the top and forget about it please.