I used to work in a clinic alongside an expert on this type of problem (I stress I'm not an expert myself).
The two types of professional who used to see children with this problem were (primarily) a SALT and a clinical psychologist (generally only if it got really bad, or there were issues about consequent slow growth, etc., which it definitely doesn't sound like is the case for your DS).
Anyway, from the CP perspective (in my unexpert recollection. BTW, this is going to sound much easier than it probably is in reality...), the general theme would (usually) be to reduce anxiety as far as possible around mealtimes. Offer a selection of foods of different textures, but never force the issue. Offer more difficult foods at times he's more hungry so more motivated. Make food, as far as possible, not an emotional event.
From the SALT perspective (in my very unexpert experience), I understand that sensory hypersensitivity in the mouth/throat and a sensitive gag reflex can become less sensitive over time with exposure and practice (which will happen naturally, on the whole).
Incidentally, I was a big "gagger" as a child. I still remember gagging (with the occasional vomit) on the skins of fruit, lumpy veg, etc.
Also when anxious. Anyway, I grew fine, was and am very happy and love my food, and basically grew out of all the gagging too (though I'm still one to have a bit of a gag if I see something REALLY disgusting, and occasionally have to try a couple of times to swallow really big tablets).
Try not to worry too much - I think this is extremely common.
I expect your GP could probably refer you if you really feel you need help (or just reassurance).