With crops like corn and tomatoes originating from the Old World and wheat and barley from the New World, were there any crops that were grown in both before the commencement of the Columbian Exchange?
I think you have the locations mixed up for maize, tomatoes, wheat, and barley.
The only common crop used between the New and Old worlds are gourds which were both independently domesticated. Gourds are great to use as vessels and water storage containers without the need to develop ceramics or make bladders from animal tissues. For the Americas, gourds seem to have been domesticated about 10,000 years ago (Smith 1997). At this time people had been living in the Americas for at least four thousand to ten thousand years. Other crops like maize and tomatoes were domesticated several thousand years later after the gourd (Coe 1994).
Coe, Sophie D. America's first cuisines. University of Texas Press, 1994.
Smith, Bruce D. "The initial domestication of Cucurbita pepo in the Americas 10,000 years ago." Science 276.5314 (1997): 932-934.
/u/Mictlantecuhtli has noted that gourds/squashes were present in both areas, and /u/Reedstilt pointed out cotton. I'll add three more: acorns and some other nuts (insofar as they're a crop, but they're edible), strawberries and beans, with a caveat on the strawberries and beans that they were different varieties.
Prior to the Columbian exchange, there were strawberries in Europe. They were tiny (10mm across was a big one), and quite intensely flavoured. The modern farmed strawberry is a cross between this species (which you can still find in woodland edges across Europe) and a strawberry species from the New World, which was bigger, but less flavourful.
Beans are also found in both Old and New World - the Old World provided broad/fava beans, and the New World provided most of the rest.
Oaks grow widely across the Northern hemisphere, and acorns have been used as food for both livestock and humans. There are a lot of sub-species and varieties unique to the Old or New Worlds, but the 'ordinary' oak, quercus quercus, is native to both North America and Europe. And as a side note, beech and hazel trees are found in both too, but are a slightly different species in each case - beech mast has been eaten by both livestock and humans, and also pressed for oil. Hazel nuts are widely eaten, although the American hazels less so. Neither oak nor beech really qualify as crops, but you could make an argument for hazel.