The Highways Agency is in general only responsible for the motorway and trunk road network, I think the general responsibility for the actual signage lies with the local 'Highways Authority', so the local county, district or unitary authority as applicable.
What has changed in recent years is that the structural safety of some road overbridges that carried unused rail formations would have been the responsibility of the British Railways Board (Residuary) - i.e. the part of the
BR▸ estate that was not transferred to Railtrack, subsequently Network Rail.
When BRBR was wound up in 2013 a new separate Highways Agency offshoot was established to deal with what is known as the 'burdensome estate' defined as "This includes legacy bridges, abutments, tunnels, cuttings, viaducts and similar properties associated with closed railway lines..." They are known as
Highways Agency Historical Railways Estate; and based in York
AIUI▸ they are effectively the maintainer of such infrastructure, but the local highways authority would be responsible for the bridge havignthe appropriate traffic signs in place.
https://www.gov.uk/government/news/brb-residuary-ltd-has-been-abolishedPaul