The legal basis for damages would stem from the local authority's duties under Part VII of the Homeless Act 1996, common law duties of care, and obligations under Article 8 of the European Convention on Human Rights as implemented by the Human Rights Act 1998, as affirmed by McDonagh, R (on the application of) v London Borough of Enfield (2018) EWHC 1287 (Admin) although in that case, there was no Article 8 infringement so no damages were awarded.
An Article 8 infringement claim was made possible by Morris v The London Borough of Newham (2002) EWHC 1262 (Admin) and R (Bernard) v Enfield London Borough Council (2002) EWHC 2282 Admin.
Following the same reasoning as the prior cases above, it is obvious that the local authority has a common law duty of care to provide a needs assessment in a reasonable time period, subject to resource constraint, under the Care Act 2014.
It also has a duty to facilitate respect for someone's private and family life. In this case, an inordinate wait for a needs assessment is highly likely to infringe on that right, causing a breach of the duty. However, what constitutes an "inordinate wait" would be a matter for the courts to determine.
The quantum (amount) of damages would also be fact-specific, but it seems likely that any award would be more than merely minimal, to reflect the fact that the statutory provision of care to someone who needs it is a fundamental part of their rights under Article 8, and unnecessary delay to facilitating those needs could be akin to humiliating or degrading treatment.