Nathan wrote:

My plan was to drill (with a pillar-drill) into the elm, ending up
about 30mm short of the outer edge, then slot the shelf onto the
supports.

I was thinking of drilling 10mm holes in the shelf, then whacking the
shelf on, but would I be better with 12mm, and podging car-body filler
in around the rods?

So that's 4 rods about 290mm long engaging with deep holes in the shelf.
You'd need to have the rods all perfectly aligned parallel with each other
to slide the shelf on using 10mm dia holes. I expect you'll need quite
generous clearance.

--
Mike Clarke