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.
Centre line of new WC is about 210mm from floor level and centre line of existing family bathroom WC is about 195mm, however as the new WC pipe drops down to the Tee piece the main soil pipe is lower than 210mm, but don't think there is very much of a fall if any towards downpipe.
The existing family bathroom WC was just a 90 degree bend from WC to soil pipe, so think this needs raising to give a better fall to downpipe.
There is a sweep tee fitted correctly to the new WC but if the fall isn't steep enough or horizontal then I guess getting backwash when flushed.
Do they make any WC pans with a higher than normal outlet pipe or maybe I install another wall hung pan but slightly higher?
I intend to contact plumber who fitted it, just wanted to see what ideas etc were available prior to speaking to him.
This photo of family bathroom was taken at original build and shows WC pipe connection.