harg
There are no signs specifically on this path. Other than an anonymous yellow arrow on a post at the North end, with no indication what it means. But this way on OSM appears to correspond to the path on the Hawthorn Trail PDF mentioned by City of London (owner of the forest) at https://www.cityoflondon.gov.uk/things-to-do/green-spaces/epping-forest/activities-in-epping-forest/walking-and-running-in-epping-forest -- which also makes it clear that those trails are suitable for walking and running. Then there's https://www.cityoflondon.gov.uk/things-to-do/green-spaces/epping-forest/activities-in-epping-forest/cycling-in-epping-forest and https://www.cityoflondon.gov.uk/things-to-do/green-spaces/epping-forest/activities-in-epping-forest/horse-riding-in-epping-forest, both of which mention "284km of shared-use paths" and point to a PDF map showing the same path. So, bicycle=yes,foot=yes,horse=yes seems correct.
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The bigger question is whether it's a bridleway. Under OSM definition "A way intended for use by horse riders (primarily)" -- no, it's NOT *primarily* intended for horses (but nor are most other bridleways in the UK). It's a shared use path mostly used by walkers and cyclists, but horses are allowed to use it too. There's no evidence (so far) that it's a PUBLIC bridleway as defined by English law.
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So my suggestion is add horse=yes, add access=permissive and *maybe* chance highway from bridleway to track.