So I’m meant to have 20mm over hang both sides. Is there a way to attach another Stringer to the existing one, to make it flush?
Yes, you will need to keep the stringer central to the newel post, so you will need to clad an equal thickness to both sides, this will mean scribing around the nosings to fit tread stringer. It is normal to have the overhang on the stringer capping, you could use a moulding on the underside to give a more decorative finish than a flat base to the capping.
SMUN has given you good advice but I'll put it bluntly, in an ideal world your newel post should be central to the baserail and handrail so the spindles end up central, if the string is off centre to the newel as you've suggested the easiest way to sort it out is to cut and scribe a false string from MDF, Ply or even softwood and glue and pin it to the existing string so that with the new thickness the newel post is more or less central. Looking at your photos you need 3/4" or 1" thickness material planted on from the living room side. Then centralise the baserail over the new thicker string, screw down and away you go.