Is the centre line of the waste outlet in the old family WC above the level of the horizontal pipe going left to new WC position. (Like Nige F's suggested Marley drwg.)
Or is it level with pipe and suffering from back wash from new WC position when that is flushed.
Apart from a one way 'cat flap' valve inserted in the pipe between old family WC and new position WC, which they don't make
some sort of height adjustment looks likely. Either raising the old wc or lowering the newly positioned WC.
Just make sure that the newly installed WC has a swept tee and that it is allowing a directed flow outwards to the down pipe.
(Swept tee's make the water flow easier in that direction, but it may have one, but the water is still backing up!)
Can you not call back the plumber who recently installed, he may have a better idea to overcome the problem.