What is the easiest way for a UK citizen to convert a non-O resident visa to a work permit in Thailand?

I'm pretty sure you can work with a type O based on minor dependent. You don't change your visa to a work permit. The work permit is not a visa. It is just certifying the right to work in Thailand. You do need a type of visa that allows you to work, and I'm pretty sure your type O based on dependent does. My type O based on Thai wife does, but it is not true for all type O reasons. You can not work with a type O based on retirement.

Either your company will provide you with documentation and you go to a neighboring country, apply for non-B at the Thai embassy there, and return. This is the most common method.

Or if your company can produce the much larger amount of paperwork and your immigration office agrees, you may be able to apply for a conversion to non-B inside the country. Most companies do not want to meet this higher burden. Also some immigration offices won't let you convert from one type of non immigration visa to another (O to B

You can’t convert a non immigrant type visa to any other non immigrant visa. You can convert an extension of stay for reason X to reason Y, but whether your new employer would be able to supply the documentation to do so is between them and the immigration office.

Most find it easier (and way less documentation needed) to travel to a nearby consulate and apply for a non B visa there.