As per ASME Sec.IX - TPI Witnessing & Certification for Welder Qualification is Not required
As per BSEN 287-1 - TPI Witnessing & Certification for Welder Qualification is required.
Summary
Witnessing & Certification of Welder Qualification depends upon the Standards used. Apart from that it depends upon the Client's requirement/Project Requirement.
If you are the individual responsible for the qualification of welders and you need to ask if a third party is required, then my response is that it would be prudent to hire a third party that is knowledgeable of the requirements of the welding standards cited.
My recommendation is to seek the services of an individual certified to AWS QC1 as a Senior Certified Welding Inspector and if you cannot find one then seek someone that is qualified to AWS QC1 as a Certified Welding Inspector.