It is still possible to source those fabrics?
Worsted 13 to 16 oz
Concerning 13 oz typical english old school cloth you can still get Harrisons Oyster or Lessers 13oz. Perhaps they lost something from the vintage variety but its close enough. Smith Woolens too. Also dugdales city and town, which is perhaps a bit less refined and HFW's Classic II which finishing is a bit more modern imo. If you can handle slightly lighter in weight schofield and smith Westminister is good cloth at 350g. Periodically H&S and Scabal do heavy wieght collections too.
Concernign 16oz cloth, options have dwindled. H lesser book is running out and does not seem to be being renewed. Harrison's P&B universal is still around, less refined than Lesser but still might do the trick depending on the pattern.
Reid & Taylor Twist
Reid and Taylor twist is no more. Stuff comes up on ebay uk every now and then (sometimes under their own name sometimes under Holland and Sherry, Scabal, and Kieth and Henderson) but mainly in 70s shades of browns and greens, the more desirable blue and greys are harder to come by. Also look for old Reid and Welsh and other defunct Scottish mills. The original stuff could be quite heavy 18oz+. This newer collection looks and feels quite nice at 13 oz :
http://www.pepperlee.co.uk/suiting/bunch-name/heritage-twist.html Also see the Halstead 3ply mohair on their site, not many patterns but the old school formula and weight.
Sportex
The old sportex is no more. Again pops up on ebay every now and then, but its mainly browns. But Dormeuil is still producing a lighter 380 g version "vintage sportex" in their British Collection. Patterns are limited but some look quite nice.
Tweed
sporting and/or Estate Tweed is still readily available. for example check out the heavy weight ranges here:
https://www.yorkshirefabric.com/collections/tweed/22-24oz
or
http://www.intweed.co.uk/goods-for-sale/heavy-weight-tweeds
H&S also sometimes has an estate tweed collection.
for donegal contact molloy and sons, also w bill. Porter and hardings thornproof collection is nice machine made donegal style suiting. also porter and hardings glen royal is beautiful softish 14 oz jacketing/suiting.
unfortunately harris and shetland are not what they used to be but the London Lounge still produces excellent shetland through lovat. But I find most of lovats standard range of tweed to be quite uninspiring though. Periodically scabal will do a shetland collection. The Marlin & Evans natural collection also looks promising.
Other Stuff
For flannel i like HFW's heavy weight stuff, fox's heavy weight stuff and Harrisons. But vintage flannel would be my preference worsted or woolen.
For linen the w bill and HFW 12/13 oz suiting is good. Would like to investigate further with direct purchases from the Irish mills.
Don't really have any recommendations for lightweight worsted current production other than mohair blends and fresco type stuff. HFW rangoon is good but no longer available, sometimes special runs pop up on SF. H Lesser vintage panama suiting is very good, and there may be some lengths left at W bills london warehouse.
The vintage goods
The vintage goods can be found sometimes for a reasonable price on ebay UK, but patience is required. Other interesting and highly desirable vintage stuff is old H&S with mink or vicuna blends. Vintage heavier weight cashmere is also something of a holy grail, harrisons and johnstons were known for this. another thing people look for in vintage cloth is old "london shrunk" markings. vintage lambswool jacketings and saxony flannels like lessers original barera collections are also top stuff. vintage lessers golden bale, especially the flannels is also highly desirable. Lots of old woodhouse, dormeuil (vintage towntex and tonik especially) and wain shiell can be great stuff too. Old italian cloth, 80s or earlier, can be great stuff too, especially for jacketing and fake tweeds and cashmere and flannels see vintage mario zegna and tons of other mills.
If your interested I have plenty of vintage brown/green twist suiting and sportex I would be willing to sell. also some heavy weight worsted, but it can be rough stuff. let me know if there is interest and will photo.
In Italy the two biggest sources for this kind of stuff that i know of is M. Cilento in Naples and il vecchio drappiere in milan. Cilento has an amazing vintage tweed selection and even gets sportex and tonik special runs made up from Dormeuil, but it will cost you. Il Vecchio is more hit or miss, but it will also cost you an arm and a leg.
Lafayette Saltiel Drapiers in Paris also has large stock of some interesting vintage often no name stuff.
Another option is to see what your tailor or any old tailor shop has in their attic or basement or just lying around.
And yes Japan remains a source but the logistics are quite complicated.