Is there a recommended way of working out the size of a joist (e.g.
50x150) required to span a 3m distance? It needs to be robust as it
will be supporting a washing machine pretty much on its center point.

One of the walls that the joist will attach to is a conventional brick
wall. To attach to this wall I was planning on cutting a grove with an
angle grinder and then using joist hangers.

At the other end, the wall is soft breeze block - so soft that I can
easily push a screwdriver blade into it. I'm assuming on this side that
I'll need to support it from ground level rather than attaching to the
breeze blocks - The new floor is only going to be about 2ft above an
existing concrete garage floor, so this would be relatively straight
forward as a solution.

Don