There is no public right-of-way across the station bridge at Sheffield (indeed, it's one of the reasons they fastidiously clear and lock the station up each night to prevent one from being implicitly created), however the council was involved with funding the bridge refurb and got assurances that it would be open to the public whenever the station was open. In the end it came down to planning permission and the council would not grant EMT permission for barriers, the secretary for state got involved and the whole idea was quietly dropped, with vague promises about looking again after the bridge just to the south of the station was replaced or upgrade.
That second bridge has now been closed for over a year. The decking was quite "bouncy" in places and had numerous temporary covers added in recent years. There was also constant evidence of vandalism (including burn marks on the decking and the metal wire which roofed it had been cut open in numerous places) and it was blind until you were on the actual bridge, so was a notorious spot for muggings. That bridge does carry a right-of-way, so I'm surprised there is no public announcements about what is planned for it.