Between Hyde Park and the river there's little cycling infrastructure but once in Battersea you could go along the Thames path or on the main road and turn off before Wandsworth roundabout.
I would recommend Shokz Openrun, not tried the Pros. Don't expect the sound to be as good as headphones but they're still very good, especially if using for spoken word. Very light, so you can almost forget they're on.
slightly used would be fine.