Unless you are under a particular contract or working for an organisation with renewal times, such as the NHS or a school, there are not time limits on a DBS check.

It is only as good as the day it is printed. There is no expiry date.

It is up to the new employer to perform a risk assessment. For example, if the DBS is over a year old, it may be prudent to perfom another check but if it was done only a week prior to the new employment, it is unlikely the applicant could have been conviced of a serious crime within that time.

It could also depend on the job role - children, security, vulnerable adults.

It is entirely up to the new employer.