I've had a similarly disgusting problem in my Renfrewshire area. Ally had nothing to do with it though.
You need a drain guy Tidy, not a plumber. It took two or three attempts to unblock mine. First was thwarted by drain guy's equipment being frozen (a couple of Januarys ago).
Next day, hosing upwards from near the street for hours loosened nothing but a pink plastic curler. Definitely not mine so it must have been stuck up there somewhere for years. The other +1 on this though was that his generator was screaming away all morning after the 'weans' next door had an all night party. Result 😄
That well meaning guy had to give up, but recommended folk with a camera on a flexi-rod-thingy. (I can't remember names, but fancy equipment means price goes up.) They revealed a fork in the pipe-work, showing that previous skooshing was straight-on, but the house was forked off to the side, which was why nothing was getting un-bunged. They could also see a lip though, so the drain from the house side was either badly built or damaged.
Finally a guy had to go up on the roof. There's a vent there at the top of the drain system. Once up, a quick skoosh (less than a second) and all was relieved. Including the guy still possibly shitting himself on the roof.
...I did have a terrible image of the roof-skoosh backfiring though and the kitchen getting pebble-dashed, but ...phew.