Drains responsibility of water company as soon as they serve 2 or more properties, I think the legislation came about around 2012 along with private pumping stations
They have to be Adopted first, under a Section 106 agreement, and if Water Company arent happy, they wont adopt. Also, as this section appears to only be taking the flow from one Property, it will remain in private ownership anyway.
OP, have you had it CCTV surveyed? This will prove without doubt if there is any physical restriction holding the effluent back, or if (as I suspect) there is a backfall. Unfortunately some Groundworkers haven't a clue about drainage, and the Heath Robinson affairs I've seen some cobble together, it's little wonder so many Folks have issues.