By the sounds of it you are low down the sewer run so you will be first to experience issues

I've had a similar situation in my first house but we had no warning signs until the sewer manhole cover outside the front door started bubbling with brown stuff on a Sunday evening

Fortunately I had a set of drain rods and, although not the most pleasant experience in my lifetime

, I managed to clear it

I've also experienced the fat and soap scum build up on our current place... All the grey water from the house drains into one drain and the u bend had sludged up... Another unpleasant experience as I had to clear it by hand with the aid of the hosepipe but all flowing freely again

Mind you, as everyone has already pointed out... If it's a rented property then it's your landlords responsibility unless you have signed a "Self-Maintaining" lease (although I didn't think they were allowed on residential property

) so give him/her a bell and make them aware that it is backing up and a potential health hazard
