Cheesemonger here. The cheese reference books I've read (Cheese Primer - Steve Jenkins in particular) say that the orange cheddar came as sort of a aesthetic marketing choice that caught on and became popular. The inspiration for orange cheddar probably comes from some styles of smoked cheddar. Cheddar is not the first style to use Annatto. Annatto is imported from Latin America as early as the 16th c, and is associated with a high quality of cheese (Like Gloucester or Leicester) and being an imported item there is a cost and status associated with it. Which as it becomes more widely available and cheaper in cost starts a trickle down effect to the upwardly mobile Victorian classes, and their more common varieties cheddar. The dutch start coloring their Gouda's with annatto around the same time period. So basically it became fashionable because it was new and imported, and separated high class cheese from your common farmers cheddar (Something almost every farm with dairy cattle would be making as a staple of their diet). This practice became more widely spread and has stuck around, even as the culture of cheese making has changed from the small farm hand made for personal consumption or small production to large factory operations.

It used to be that cows which grazed on higher quality grass (possibly also certain field flowers) produced a more yellowish milk than those grazing on hay or in bad grass fields. I think this has something to do with a higher beta carotene intake, if I remember correctly. So the milk made from summer/spring cow's milk from cows fed a higher quality diet tasted better and was a different colour than cows milk from winter or low quality grazing fields. This meant cheddar cheese from these cows had a better flavour, and was a higher quality and a higher price. People could distinguish the quality of cheese on sight because of the yellow-orange colour, which is why lower quality cheese makers started dying their own white cheese an orange colour. As far as I know, they weren't the same shade of orange we now associate with cheddar cheese, it was more yellowee. I would imagine it was hard to recreate without fucking up the taste, and so whatever makes cheddar really orange must both be a strong pigment without much flavour, and that's why it's chosen.

Today most cows are not grazing on the same kinds of grasses, and are treated differently by mass dairy farms so we don't get the same yellowish milk that we did when cheddar cheese first became popular. However the habit of dying it orange did remain because that's what people were used to seeing.