I get all my auger bits from 'UK Drills'. The have 15 diameters, and come in lengths of 155, 235, 460mm, and some are 600mm. They're the style with straight-sided flutes that don't wander off:
Auger Bits
I also by my Forstner bits from there. The have round shafts, come in 23 sizes from 10 - 60mm diameter. I mostly use them either in a drill press or lathe, especially for holes that are of any depth.
Forstner Bits TCT
They also stock a wide range of other drills, and despite the low prices, I've found the quality and durability to be excellent.
As for hand braces and auger bits that were used with them, I'm secretary of a woodturning Club and for many years we've supported a charity called Tools with a Mission ('TWAM') who collect old tools, refurbish them, put them into kits and send them out to Africa, where they train people in a range of skills and equip them with kits of tools so they can be self-sufficient, earn a living, support their families and send their kids to school. They also send out manual and electric sewing machines and teach men tailoring skills, and women dressmaking.
They have collectors in various part of the UK and the tools go to a central refurbishing centre at Ipswich. Last year they sent 14 shipping containers with 10,000 toolkits to four African countries. Carpenters, mechanics, electricians, bricklaying, groundwork etc.
Who We Are | Tools with a Mission
We often get hand braces and auger bits donated.
The braces tend to all be the same size. The picture attached below is what one Club member donated from a garage clear-out, at our recent meeting. Along with a brace and fifteen augers, there were more than fifty files and rasps which, though rusty and mostly unhandled, had had little use. We've also donated power tools - two lathes, woodturning tools, routers, scrollsaws, bandsaws circular saws, as well as handsaws. Often donated by people clearing out workshops of deceased relatives. Also, three electric sewing machines from members wives who had upgraded to newer models.
Hope that's of interest.