Is it the designers’ responsibility to inform clients of the need for a party wall agreement?

The designer does not have any responsibility to inform the client of any aspect of the Party Wall Act, unless specifically referenced in the original quote and brief. Having said that, it is good practice for the designer to advise the client of any other services that they may need arising from the design and construction of the project. You may wish to consult a surveyor who specialises in writing party wall agreements for further information.