If I am a Nominet TAG holder, technically, can I run my own WHOIS privacy service by registering domains in my companies name on behalf of the client? Or do I need to take extra steps?
No reason for clients to not be Registrants. Just use your company details as contact for the privacy service.
https://registrars.nominet.uk/namespace/uk/privacy-service-framework tells you what you need and the EPP flags
Quite right, @Edwin I just created a new thread on the forum before spotting this thread (and your response) https://www.theregister.co.uk/2018/04/14/whois_icann_gdpr_europe/