You can usually turn the bottle upside down and stand it on its lid. So once you get to the point where there's a little remaining ketchup/mayonnaise and it won't come out, just store the bottle upside down. After a few hours, any remaining ketchup/mayonnaise will have fallen down to the lid. Then the next time you need ketchup/mayonnaise, you can carefully open the lid and ketchup/mayonnaise will easily come out.
If for some reason the ketchup/mayonnaise doesn't fall down to the lid using this method, then I would recommend that you get a chopstick and insert that into the bottle. It will be thin enough to fit in the thinnest bottle necks, and should be long enough for you to reach the bottom of the bottle. Then hold the bottle almost upside down and use the chopstick to scrape the ketchup/mayonnaise from the bottom and have it fall down near the lid.