The most-recent Race podcast focused on Newey's appointment (condensed version
here) think that Christian Horner may still end up in AMR, with the likeliest position as Group CEO. Andy Cowell was both Group CEO and Team Principal and with Cowell moving over to work with Honda and ARAMCO on the powerplant side and Newey taking the TP role, the Group CEO role is currently open.
Under AMR's current management structure, Newey as Managing Technical Partner was effectively equal with Cowell, so one would expect Horner as Group CEO would be the same with Newey, now that he holds both the TP and MTP roles (and is an equity holder in the team). This would allow Newey to focus on the technical and development side of the chassis while Horner would handle the team and sponsor management roles (something he is admittedly very good at).