It's not "highway" but a foot/cycle path alongside the bridge. The West path has been closed for quite a long time, it's the East one that's now been closed as well.
It is not marked on the OS maps as a distinct right of way, so I believe it to be considered a cycleway (with foot use permitted) as part of the highway. Remember that legally a highway can be comprised of a carriageway, cycleway and footway.
I suspect the Humber Bridge Acts give the board the power to take short term emergency management actions but I have not found a copy of the 1959 or 1971 Acts online.
The 2013 Act allows them to "do any other thing (including the provision of facilities and the carrying on of any business) which in the opinion of the Board can usefully be undertaken by the Board by virtue of the Board's functions in relation to the bridge and which, in the opinion of the Board, is likely to provide economic, social, cultural or environmental benefits to all or some of the population of, or visitors to, the Humber area." They may be relying on that power, but it being a net benefit seems a bit shaky if the closure continues into the working week!
Later in the 2013 Act, it says "The Board is deemed to be the traffic authority for the highway on the bridge for the purposes of the Road Traffic Regulation Act 1984" so they can make Traffic Regulation Orders to ban bikes and foot passengers, but I think it would take longer and be a difficult argument, especially as the Act states later that they must be allowed to cross with nil toll.