Based on a description from the RSPB, foot=permissive and everything else private seems the most appropriate.
There is very limited horse permit access, but it's limited enough so that "private" applies there too.
I've also taken the liberty of updating the "accessible" track following the RSPB's email.

To discuss these changes, please email data@openstreetmap.org with a subject line of "[Ticket#2024070810000311] Cycling at RSPB Broadwater Warren Nature Reserve"