As far as I'm aware, there is no legal requirement to publish any personal information, although failure to do so is clearly at odds with all best practice guidelines for a representative of a local community. The ICO Model Publication Scheme includes the following:
"Class 1 - Who we are and what we do
(Organisational information, structures, locations and contacts)
This will be current information only.
N.B. Councils should already be publishing as much information as possible about how they can be contacted.
Who’s who on the Council and its Committees
Contact details for Parish Clerk and Council members (named contacts where possible with telephone number and email address (if used))"
Slightly ambiguous, but it certainly doesn't say you must.
New regulations last year allowed candidates to withhold all of their personal information throughout the election process for the first time, as part of measures to counter intimidation and violence, so a candidate's address could be replaced with "Resident of {district name}" or similar.