Basic Economy on Delta (and the other US carriers) is generally not combinable with any fare that is not basic economy. If you look up a DL basic economy fare in ITA Matrix or Expert Flyer, the fare rules will tell you such.

British Airways does allow you to combine Basic fares with other economy fares as well as Premium Economy, Business, and First. I checked Virgin Atlantic and they do not seem to allow combinations like that. One-ways to/from Europe are ridiculously expensive, so if you want to book Delta, you will probably have to pay the upcharge from Basic Economy to Main Cabin on that segment (probably about $70).