is there a time limit as to when a local authority has to start a home assessment?

Answer 1

1. I am assuming that you mean the local authority's social services department carrying out an assessment of an individual's social care needs. In that case, all local authorities are supposed to follow Department of Health guidelines - government inspectors will report on how well a local authority adheres to the guidelines. Here is the relevant page.
http://www.dh.gov.uk/PublicationsAndStatistics/Publications/PublicationsPolicyAndGuidance/PublicationsPolicyAndGuidanceArticle/fs/en?CONTENT_ID=4009653&chk=nadbwI

2. The local authority itself will have information on its care services, and this may well include the turnaround times that it has set itself.

Answer 2

This depends on from what point the assessment is started. In my authority, the assessment starts from the point of the initial phone call/letter. The duty officer will assess the priority and clietns usually go onto a waiting list. Too few staff to visit usually.