guv wrote:
What I would like to know is if this drain is actually my
responsibility given its on y property?

It depends where the blockage is - not where the manhole is. The manhole
in your garden is almost certainly your responsibility.

There is probably a physical obstruction of the pipe and trying to push
down from the flooded manhole will probably only further compact the
blockage. The best course of action is probably to find the manhole
downstream, which should be dry, and poke rods up till it hits blockage.
That will give you the approximate location of the blockage and you can
take it from there.

If using rods, do not let them become disconnected in the pipe.

Owain