What aircraft could make non-stop transatlantic flights in 1958?
The first scheduled trans-atlantic air service was made possible with Zeppelins. The LZ-127 started flying the route in 1931, mostly to Lakehurst, NJ, or Pernambuco in Brazil, but was disassembled in 1940.

The first airliner which was able to cover the distance (back then it was Berlin - New York) was the Focke-Wulf 200, which in 1938 needed almost 25 hours East-West and 20 hours on the return leg. With its slow cruise speed, wind made a lot of difference. Sadly, the US did not allow Lufthansa to start the scheduled service it had planned to open in 1939. Lufthansa had opened regular air mail services in 1934 between Europe and South America using flying boats and a relay ship in the middle of the Atlantic. Air France added its transatlantic mail service flying from Africa to South America and back.

The first heavier-than-air transatlantic passenger service opened between New York and Marseille in 1939 on Boeing 314 Clippers operated by Pan Am. It crossed the Atlantic between Foynes, Ireland and Botwood, Newfoundland. A British service, also using flying boats, was in preparation but stopped by the beginning of WW II.

In the 1950s, there were already several options for transatlantic air travel, but none of them non-stop from Chicago: Pan Am, TWA, Trans Canada Airlines (TCA), BOAC (successor of Imperial Airways), and Air France all operated large propeller aircraft which covered the distance with refueling steps, typically in Gander (Newfoundland) and Shannon (Ireland). Typical aircraft were the Douglas DC-4 and DC-6, and the Lockheed 149 Constellation, but also the French Latecoere 631 flying boat was used until 1955. Of these airlines, only Pan Am and BOAC serviced O'Hare in 1957, before the first international terminal opened there in the following year.

Still, the prevailing eastward wind made the East-West crossing hard and required airlines to reduce payload, and only the advent of faster  airliners like the Douglas DC-7 (from 1956) and the Lockheed 1649 Starliner (from 1957) made the transatlantic services more reliable and economical. Beginning with the DC-7C, their range was sufficient to support transatlantic service directly from Chicago O'Hare.

In October 1958 the first transatlantic jet airliner service was opened by BOAC using the de Havilland Comet 4, followed shortly after by Pan Am, operating the Boeing 707 and connecting London with New York (with an occasional refueling stop at Gander on westbound flights). O'Hare could only participate in this progress after its main runway was extended to accommodate large jets in June 1960.

Therefore, the most likely airplanes to fly transatlantic directly from O'Hare would have been the DC-7C (but operated by Pan Am, not TWA), the Lockheed L-1049G and the L-1649, which both were operated by TWA. For the flying time, the 3628 miles from JFK/Idlewild to Paris Orly took 14¼ hours with the DC-7C and 14.83 hours with the Lockheed 1649. So it is fair to assume a flying time of 15½ to 16 hours for the slightly longer distance between Heathrow and O'Hare.

Quote from Wikipedia:


  In January 1958 Pan American scheduled the DC-7C from Orly to Idlewild
  in 14 hr 15 min; TWA scheduled the 1649 in 14 hr 50 min.