It took me ages to find this....
There is no set time limit for carrying out an assessment in the Care Act 2014 or in the Statutory Guidance. However, paragraph 6.29 of the Care And Support Statutory Guidance provides:-
6.29 An assessment should be carried out over an appropriate and reasonable timescale taking into account the urgency of needs and a consideration of any fluctuation in those needs. Local authorities should inform the individual of an indicative timescale over which their assessment will be conducted and keep the person informed throughout the assessment process.
This means that assessment should be completed, which should include providing a report or record of the outcome of the assessment within a timeframe appropriate to the needs of the applicant. This could give rise to a very short timescale in an urgent case.
Where support is needed urgently councils can be required to provide this before completion of an assessment - see the case of R (AA) v Lambeth LBC  EWHC Admin 741, QBD. That case was before the Care Act 2014 was introduced but Forbes J stated at ‑
"Nothing shall prevent an authority from temporarily providing or arranging the provision of community care services for any person without carrying out a prior assessment of needs if, in the opinion of the authority, the condition of that person is such that he or she requires those services as a matter of urgency"
Section 19(3) of the Care Act 2014 contains a wider power to provide services in urgent cases, before completing an assessment but it is still contingent on the local authority forming the view that the situation is urgent; therefore, the overarching power of the High Court to grant interim relief remains important. It states:
19 (3) A local authority may meet an adult’s needs for care and support which appear to it to be urgent (regardless of whether the adult is ordinarily resident in its area) without having yet—
(a)carried out a needs assessment or a financial assessment, or
(b)made a determination under section 13(1).