Hi and welcome to the forum.

The sizing of the screw depends on three things.. Firstly the speed you plan to rotate it and how its physically mounted - too thin a screw will 'whip' at speed. on an 800mm length of 20mm fixed in bearings both ends the critical speed is 2700rpm, thats 13.5m/min traverse on a 5mm pitch - I dont think that an issue!

Secondly, what is known as Euler buckling... the tendency of the screw to bend under the load (think of it as a column with a weight on the top). Again for a 800mm long, 20mm screw in a fixed bearing at each end held in tension the load is around 7000N (7 tons), again not an issue for a mill of this size (max cutting force probably a tenth of that)

The thrid factor is simply the load rating of the ballscrew and nut... can it transmit enough force through the thread of the screw.. looking at a typical manufacturers catalog (Nook), a 20mm x 5mm nut can handle 10800N dynamic load for 1million rotations or 21000N static thrust. The screw will buckle long before the nut fails!

I'd go with the 20mm...