Chris,
According to the publicity sheets for the Z and Z1 models, the wheels are listed as 26" with 26 x 3.5" tyres. Now, this is the prewar way of measuring wheels and tyres, 26" overall diameter x the tyre section(3.5") - so the rims are acually 26"less twice the tyre section (7"), in other words, 19". Postwar, they were known as WM2 x 19" rims. WM3 rims were usually reserved for use with 4.00 tyres.
The publicity sheet also lists the spokes as 9 gauge. There is no mention of the spoke pattern, but I would expect 'cross 3' to be the norm.
Hope this helps,
Regards,
Eddie.
P.S. the spoke pattern will look considerably different to that shown in your illustration, as all spokes on Douglas wheels go to the spool hub, rather than the brake drum.