It depends on what your intention is. A 100% correct bike - tricky thing. Something from the 1950s can be tricky, but doable. Something from, say, the 1920s or 30s will be quite hard to get a complete, period-correct bike. If you're in it for a rider with some adaptations, then it's doable. Pre-WWII bikes tend to be tougher to get parts for than post-war bikes because so much stuff was scrapped or damaged during the war.

My recommendation for something at the "easier" end of the spectrum would be a 1940s or 50s Raleigh, BSA, Phillips, or Hercules road bike. They're fairly well-made, were produced in pretty good numbers, not overly expensive (usually), and the parts can still be found on eBay and similar places. There is also support available through the Veteran Cycle Club out of the UK for those kinds of bikes. There is a growing number of clubs online through Facebook and similar places. Frame and fender graphics are available for many brands through H. Lloyd in England.

A notch more difficult would be one of the better Schwinn bikes - any of the fillet brazed models or the Paramount. Pre-1960 Paramounts tend to be on the expensive side, but fillet brazed models like the pre-war Superior or the 1940s Continental (not to be confused with the welded 10-speed from the 1960s and later) are still available. These bikes tend to follow English-style parts and designs. People are often surprised to see that Schwinn made some very good touring and road bikes for adults right before and right after WWII. If you want to go really esoteric, Schwinn even made clubmen to compete with the British bikes.

The French and Italians also made some very good bikes in that period, but I've always found them a little more difficult to locate parts. But certainly if you find one, they're worth a look. I have a soft spot for Automoto and Gnome Rhone bikes from this period.