At the time the war broke out, it probably didn't matter much to the factory owner from a business perspective. Although the Confederacy hoped (and counted on) European demand for cotton would force those countries to recognize and support their cause, bumper crops in 1860-61 meant that England enjoyed a large surplus of cotton. The economic distress caused by disrupted Southern cotton exports mostly wouldn't be felt in England until mid-1862 or so.

Whom the owner would support would likely be personal, and truth be told I don't know where factory owners themselves tended to fall. Some British cotton merchants, though, made statements that they wouldn't allow rising cotton prices to push them into supporting a cause they considered repugnant (many saw supporting the Confederacy to be the same as supporting slavery). British textile workers, by and large, supported the Union throughout the war, even when the ill-effects of disrupted cotton supply hurt them economically.

What were those effects? Well, as you might imagine, work slowed considerably, leading to widespread unemployment or at least reduced hours. In some places, notably Lancashire, this led to real unrest and suffering. But these things did not by and large lead workers toward support for the Confederacy. For one thing, cotton continued to flow into England (at reduced volumes) thanks to Confederate blockade running and growing imports from places like Brazil, Egypt, and India. This kept many textile mills running, if just on a part-time basis. Charity relief further helped alleviate economic suffering for British workers. Finally, many workers seemed to associate the emancipation and enfranchisement of American slaves with their own efforts at labor reform within England, and therefore supported the Union.

Source: Howard Jones' Blue and Gray Diplomacy