(Review requested)

The outline of the path looks fine, as it's from your own GPX file.

However you probably don't need to add an access=no tag here. If horses and bicycles are prohibited and it's pedestrian only, you could additionally tag it with horse=no and bicycle=no. Alternatively, you could replace highway=path with highway=footway and dispense with the access tags entirely. (NB This is just my opinion, what you have mapped isn't wrong and won't cause any problems for data consumers and you don't need to change anything.)

You might find the following useful
https://wiki.openstreetmap.org/wiki/Tag:highway%3Dpath
https://wiki.openstreetmap.org/wiki/OSM_tags_for_routing/Access_restrictions#United_Kingdom
https://wiki.openstreetmap.org/wiki/Access_provisions_in_the_United_Kingdom