…I’m all of a quandry about how I get them out of the ground…
I’d be surprised if the asbestos went into the ground. Chances are the underground pipes will be clay.
From the photos it looks like there is a clay pipe socket is sticking up out of the concrete, with the end of the down pipe sealed to it with mortar. I would expect that you could pull the down pipe out of this joint fairly easily and then carefully break out the mortar, taking care not to crack the clay pipe, or drop rubbish down the ‘ole. 2nd photo looks like there might be a short length of clay pipe standing inside the socket in the concrete, with the down pipe in the top of that.
There may or may not be a ‘gully’ (like a u bend) under the concrete from each connection. FWIW, I would try to leave the drains where they are, (assuming you’re not taking the concrete up anyway).
You can fit a couple of 135* bends to step the bottom of the down pipe back towards the drain (you can run the down pipe slightly off the original line to help with this if you’re getting new guttering anyway).
If you are lucky, something like this will fit neatly in the clay socket, if not, you could use a clay to 110mm adaptor, then a 68mm downpipe to 110mm adaptor, but it might look a bit crap.
I am not a plumber.