Hello and welcome to the forum. I had Westies for many years so I know the breed very well. As you probably know, they have a reputation for eczema and one of mine had this but only mildly. It responded to treatment.
Well done for researching dog food products and yes, Bakers has a poor reputation and low score on the Dog Food Directory. You don't say which of the Natures Menu products you have chosen. I assume it is one of the tins or pouches? Normally we advise people to transition from one food to another over a period of several days. I think you should stop the treats, Caesar etc and transition from the Bakers itself. You won't need the joint or dental stick. NM has all your dog needs and his dental hygiene will be improved by using their product. These treats are, like Bakers poor quality. If you like giving treats, choose a really good quality one. Mine have sea jerky (dried fish skin). They are very crunchy and help to clean their teeth. They are also low calorie. Fish4Dogs do them but you might be able to find it in your local pet store.
You might find that your dog's poo may become a little bit hard. If this is the case, you can add a little mashed sweet potato or similar to provide bulk. It will take several weeks (probably at least six) before you see a difference but eventually your dog's coat, breath and general liveliness should be noticeably better. The pink staining on his coat might also reduce.
Don't forget that you can always telephone Natures Menu for advice about their products - they will be able to guide you through the transition. You have chosen a good product and I am hopeful that your little dog will soon be much better in every way.