I agree its more than possible that all your foul waste including that from your gulley heads towards the street and in that case you wouldn't have or need manholes in the back garden.
Where does the waste from your bathroom go? Into the same gulley or somewhere else.
If you lift the cover as suggested and then get someone to operate toilet, run taps in bathroom, kitchen etc sequentially you will start to see what happens with each. You should find that there will be a separate outlet into the manhole for each source. If water runs freely through manhole from eveything except kitchen then thats where the problem lies and you will have identified which is the kitchen gulley because it will be the one through which no water flows.
I'm inclined to think a blockage at the gulley outlet which should have a trap. Roll your sleeve right up, peg on nose just in case and get in there and ferret about. Looked at a gulley trap not so long ago which wasn't running freely. It was full of sand and a few unmentionables!! Hoiked all that out and off it went again.
If that doesn't work (ie the blockage is farther down) then you'll probably have to rod it and most likely from the other end because you won't get the rod round the gulley trap.
IF all else fails call in a drainage contractor. He may suggest sticking a camera down there to find out whats going on.
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