You don't need to visit Graham Browne just because you want classic fabrics. I daresay that most tailors will have the big names (Holland & Sherry, Harrison's etc). In any case, I think the priority for the OP is to get a well-cut suit (with fabric playing an important, though slightly secondary role). In addition, if a first suit is a nice shade of navy or charcoal, most, if not all tailoring houses will have something they can offer.
As I've reiterated, I've only seen a couple of suits coming out from Graham Browne that would pass muster, and my poor experience there has changed my opinion on them. The adage is true -- if it's (Graham Browne's prices) too good to be true, it probably isn't. If, however, the OP has had some experience bespeaking suits, and has confidence in directing the tailor, then you might consider using Browne, though there are many tailors better than him at marginally higher prices.
I actually meant never trust a UK tailor to make you a suit that will have the Italian vibe, I have not met a UK tailor that can handle paper weight stuff well yet, in the other side of the world, I think Hong Kong tailors are very good with light fabrics. anyway, thanks for the input.
I have tried Graham Browne myself, and I can say their garment making is top notch However I don't think their pattern cutting skills is on par with the big names, hence the price difference, plus you better be specific of the fit e.g. shoulder width and length with GB, they don't give much constructive advices.