I don't think you run much risk of a fake vintage uke, unless you're looking at the very few which sell for big money. The majority of vintage ukes go, in the UK, for between £200 and £600, some for even less. The amount of work needed to fake one makes it uneconomic! There are occasional high-price ukes - a Martin 5k is many times more expensive, for example, but there aren't many of those around.
Most vintage ukes in the UK are sopranos - if you're looking for an old tenor you might be looking for some time.
On pricing, a shop will want more than a private seller, because it has to make its margin and cover cost of capital if the uke doesn't sell for some time.
I've concentrated on the UK market - lots of ukes for sale in the US, but Customs duty and VAT will make them less attractive than they seem.
I'd say the main possibilities are:
Mahogany (20s-60s). The Martin Styles 0 and 1 are most common, with the 0s going £300-450 and the 1s nearer the top end. I've never heard someone dislike a Martin. A "The" Gibson uke is also good - rarer than Martins, some like them more, some less. Both Martin and Gibson made top end models whose prices go into the thousands - you'd have to research these further if you wanted to spend that much. But a Martin Style 0 is unlikely to be a bad buy. Other names include Favilla - very rare in the UK, about 2/3 the price of Martin and many say as good. If you find one and don't want it let me know! Below the £200 price mark are plenty of ukes by lesser-known makers, or unidentifed - some of these are good, some not, you can only tell by playing them.
Koa, Hawaiian made. Kumalae is the most common name here, and there are plenty in the UK - they seem more plentiful than in the US. £150-£350 depending on decoration and condition. You may also see Royal Hawaiian, which is the name of a hotel in Hawaii. Most ukes with this brand were made by Kumalae, but perhaps not all. These Island ukes are completely different from a modern uke - narrower neck, frets set direct into neck, very light weight. There are quite a lot of "tourist" ukes, often with a palm tree or hula girl painted on - these are wall decoration, not for playing!
US mass-market imports. For example, the Roy Smeck Harmony ukes - mahogany body, some with plastic fretboard, some with wood. These go under £200. Decent players and a bit of history, but not fine instruments. Other brands (similarly priced) include Stella, Supertone. Regal.
Banjo ukes. Lots of these around, selling for between £100 and £200 as a rule. If you buy one with a reasonable-sized pot (say 8 inches) which uses tension hooks rather than screws to adjust the head tension, it will probably be restorable to become a reasonable player. Hundreds of brand names here, so hard to pick out two or three as the best to look for. If you want a BU, study the eBay listings for at least a month and research the names which come up. John T's banjo shop (findable via Google) will supply parts for restoration.
Finally there are all sorts of unusual/rare ukes - plastic ones (Maccaferi, Flamingo, Islander, Beatles/[insertband name]/Mickey Mouse), camp ukes (circular body), Stroviol (with a gramophone horn instead of a body, the Roy Smeck Vita, Lyon & Healey Shrine, etc. If you find one, research, research.
The most important thing with a vintage uke is its condition. Scuffs etc don't matter from a playing point of view, but cracks and splits do. Most are repairable, but at a cost. Some are not. Repaired cracks are OK, if well repaired, but if not could be unfixable (if for example someone just epoxied the crack shut). My Kumalae, which I adore, has no cracks at all and sounds completely wonderful. I might have settled for a cracked/repaired one, but would want to play it before buying.
So, the advice bit:
1. If you want a uke that's like your Kala/Ohana/whatever, but older and nicer sounding, buy a Martin/Gibson/Favilla for £300-600.
2. If you want something completely different, buy a Kumalae for £150-350.
3. If you want a playable piece of history, buy a Roy Smeck Harmony for under £150.
4. If you want to upset the neighbours, buy a banjo uke.
What you could do is sign up to Ukulele Cosmos, most of whose members are in the UK, and post a wanted message: "I have up to £X to spend on a vintage uke - what have you got?" You might get a good deal.
[Edit: you posted your price range since I started replying. For £50 you get something anonymous (if you can find anything). £100-£200 gets you a banjo uke. £150-£200 gets you a repaired Kumalae, though it might take you a year to find it. For US posters, Kumalaes are about 2/3 the price in the UK that they go for in the US!]
