What are the rules and regulations about employing overseas nursing staff when setting up a nursing agency and do I need to register with the CQC?
The company will need to apply for a Skilled Worker Sponsor Licence to sponsor skilled non-UK nationals to work in the UK (this could include Care Workers, for example). The process involves completing an online application form and providing mandatory information and documents to the Home Office. One of the mandatory documents for care organisations is their CQC registration so, yes, they will need to register before applying for the Sponsor Licence.
In applying for the Sponsor Licence, the company will need to demonstrate that it can comply with various duties (i.e. rules and regulations). There is more information available here: https://www.gov.uk/government/collections/sponsorship-information-for-employers-and-educators