In my opinion the LA would be deemed the 'client' and therefore CDM Regs, HSWA, etc would apply. In the case of refurbishment work an asbestos D/R survey would be required along with any other relevant pre-construction information for the PC.

L143 Regulation 4 Duty to manage asbestos in none- domestic premises.
This regulation covers the duty to manage asbestos in non-domestic premises. It requires duty holders to identify the location and condition of asbestos in non-domestic premises and to manage the risk to prevent harm to anyone who works on the building or to building occupants. It also explains what is required of people who have a duty to co-operate with the main duty holder to enable them to comply with the regulation. Non-domestic premises includes the common parts of domestic premises.
However, the requirements do apply to common parts of premises, including housing developments and blocks of flats, but do not place any direct duties on landlords for individual houses or flats.
Examples of common parts would include foyers, corridors, lifts and lift shafts, staircases, boiler houses, vertical risers, gardens, yards and outhouses.
The requirements do not apply to rooms within a private residence which are shared by more than one household, such as bathrooms, kitchens etc. in shared houses and communal dining rooms and lounges in sheltered accommodation

L143 Regulation 5 Identification of the presence of asbestos
This regulation requires employers to identify the presence of asbestos and its type and condition before any building, maintenance, demolition or other work, liable to disturb asbestos, begins. It also sets out the requirement to arrange a survey if existing information on the presence of asbestos in the premises is incomplete or appears unreliable

CAR 2012 applies to all work with asbestos.
It requires employers to prevent exposure of employees to asbestos or, where this is not reasonably practicable, to reduce exposure as low as is reasonably practicable.
The spread of asbestos should also be prevented
CDM Requires the client to pass on information about the state or condition of any premises (including the presence of hazardous materials such as asbestos) to the CDM Coordinator before any work begins and to ensure that the health and safety file is available for inspection by anyone who needs the information

The client can be the beneficiary of the work, not necessarily the finance source.
Different councils do this differently - some 'take over' the work, and COULD be considered as the client. Others just give the money - leaving the domestic client in charge of the work and process.
I have asked this specific question of HSE CDM specialist inspectors - who is the client? - to get the response: 'Hmm. Grey area'.
Any contractor would have to consider the asbestos risk for work in any property built before 2000, if it needs a surveyor to check, that's part of the price for the job.
A pragmatic approach recommended: purpose of regulation - protect people.
Have appropriate controls specified, and a survey if necessary, but don't get carried away with CDM detail, which adds little, if any, additional safety.

Grant aided does not necessarily mean LA paying 100% of the contract sum.