In general, yes, but not if the employee holds a 'politically restricted post', as defined by Section 2 of the Local Government and Housing Act 1989. This section defines a list of posts within a local authority which are always politically restricted:
The following persons are to be regarded for the purposes of this Part as holding politically restricted posts under a local authority—
a. the person designated under section 4 below as the head of the authority’s paid service;
b. the statutory chief officers;
c. a non-statutory chief officer;
d. a deputy chief officer;
e. the monitoring officer designated under section 5 below;
f. any person holding a post to which he was appointed in pursuance
of section 9 below; and
g. any person not falling within paragraphs (a) to (f) above whose
post is for the time being specified by the authority in a list
maintained in accordance with subsection (2) below and any directions
under section 3 [F1or 3A] below or with section 100G(2) of the Local
Government Act 1972 or section 50G(2) of the Local Government
(Scotland) Act 1973 (list of officers to whom powers are delegated).
It further requires local authorities to draw up a list (subsection 2) of further politically restricted posts if they fall into certain job categories:
The duties of a post under a local authority fall within this
subsection if they consist in or involve one or both of the following,
that is to say—
a. giving advice on a regular basis to the authority themselves, to
any committee or sub-committee of the authority or to any joint
committee on which the authority are represented, or, where the
authority are operating executive arrangements, to the executive of
the authority, to any committee of that executive, or to any member of
that executive who is also a member of the authority];
b. speaking on behalf of the authority on a regular basis to
journalists or broadcasters.
The restrictions on these posts are defined by The Local Government Officers (Political Restrictions) Regulations 1990, and include restrictions on running for office, acting as an election agent, performing certain functions in a political party, canvassing for a political party, speaking publicly "with the apparent intention of affecting public support for a political party", or publishing any written or artistic work which could affect support for a political party.
In the particular video you reference, it could be argued that if the council employee held a politically restricted post, she could have been in breach of paragraph 6 of Schedule I:
- The appointee shall not speak to the public at large or to a section of the public with the apparent intention of affecting public
support for a political party.
Even if not holding a politically restricted post, paragraph 9.3 of part 5 of the Constitution of Hillingdon Council (the local authority encompassing Uxbridge and South Ruislip) states:
For other officers, not subject to these restrictions, it is
important that you do not carry out any political activity
in a way that might lead the public to think you are
acting as a Council employee which could represent a conflict
of interest with the stated aims of the Council.