As a continuation of my last post, a trip to London should include a visit to Jermyn Street shirtmakers - Hilditch & Key, Harvie & Hudson, Turnbull & Asser being the most well-known. I found this list of London shirtmakers to be fairly comprehensive. Personally, I would also try to make it over to Budd Shirtmakers, which, if I'm not mistaken, is the only remaining London shirtmaker that has all of its workrooms on the premises. Budd is known for its formal shirts, both black and white tie.
For jackets and suits, one really has to pay a visit to Savile Row and get fitted for a handmade, bespoke garment. Although many of the larger houses make visits to the US, I would prefer to get measured up, select fabrics and get a feel for a tailoring house at the premises itself. The key question for any first-time customer is which tailor? The Row is home to several very well-known houses including Anderson & Sheppard (no website still!), Henry Poole, Huntsman & Sons, Kilgour and Gieves & Hawkes.
An alternative is to investigate Savile Row-trained and apprenticed tailors who are now working independently (or smaller tailoring houses such as Dege & Skinner). Tailors such as Thomas Mahon are generally more affordable and may be a better choice for a first experiment in bespoke tailoring.
Unless you have fairly deep pockets (presumably handmade - lol), my advice is to do your research thoroughly. Pay a visit to the establishment or individual tailor and talk to existing customers. There is a host of other considerations for anyone embarking on a first bespoke suit or jacket, probably a topic I will cover in a later entry.