Linda, you dont actually state what your company does, other than that your Engineers undertake welding activities that are on safety critical items of plant.
Like Bob Youel states, if your engineers are not coded, I would also advise you to stop the process and arrange for the necessary training & testing required to achieve coded status.
As to what code it rather depends on the relevant industry & standards that are specified. In mine, Oil & Gas & Energy related, it is generally ASME IX & BS EN 288 supported by relevant client specifications, and witnessed by an independant 3rd Party Inspection process, generally Lloyds, DNV, Plant Safety or whatever has been agreed with the client.
In many fabrication companies, the first step is to get the weld procedure itself qualified, generally this will consist of the relevant material (Pipe to pipe, Plate to Pipe, Plate to Plate, T-Piece or whatever is specified by the standards) surpaces being prepared, Non Destructive Tested (ie Dye Penetrant Inspection on Non Ferrous Materials, Magnetic Particle Inspection on Ferrous Materials), usually welded together by the company's best welder (in order to obtain the relevant parameters such as weld speed, heat input, pre-heat, etc). The weld itself is then subjected to relevant NDT again, Dye Pen, MPI, and usually Ultrasonic Inspection and often X-Ray (dependant on standards/client requirements). The test piece is then sent off to an independant test lab for destructive testing (charpy impacts, hardness, tensile tests, weld x-section macro/micro photographs etc) all witnessed by company & independent inspection authority. At that point if all testing passes the procedure is then approved. The welder is qualified and a separate qualification is then approved by the Independant Inspection Authority. Adiitional welders requiring qualification must then use the same parameters specified within the qualified procedure when constructing their test piece, which is only subjected to NDT inspection only - No additional destructive testing is required.
A point to bear in mind, the welders can then only be qualified to that procedure within the essential variables specified by the relevant standard used. If an essential variable changes ie material type, welding wire, heat input, material thickness, pre-heat temperatures, post weld heat treatment weld position, etc then a new procedure must be raised and thus new welder qualifications.
Generally a qualification is only valid for 6months on the basis that the procedure has been followed regularly. It can be re-approved by an inhouse employee on the basis they hold the appropriate qualifications (ie PCN/CSWIP welding inspector), but must be re-approved at least annually by the Independant Inspection Authority.
One thing I can say is there are so many variables to watch out for, as such it is a specialised subject, which can bring its own problems, especially within the material properties/metallurgy aspect.
Sorry for being a little long winded, but as various posters state, it is often best to seek external expertise where applicable - the welding institute, welding wire manufacturers (Esab, Phillips, Oerlikon, etc), your own suppliers of weld consumables/equipment, etc are very often able to assist.
Alex