I mostly hear 增添 in most contexts, but lately have come across 新增 and 添加. What are the differences in tone and register?
These words are very similar in meaning. There are some more variants like 添加，加添，添增……
新增 is generally used in the sense of "add a new feature", often in the context of software. However, it can also substitute for other variants of "add". It puts more emphasis on the fact that something has changed, rather than what has been added. 新增 is often used as an adjective.
增加 tends to be used with things that can be expressed in numbers, like money, weight and so on. 增添 is more commonly used for more abstract things like content or effort. However, it's not wrong to switch them around.
添加 seems a bit more formal, more likely to appear in written language. It's often used for sentences like
牙膏中添加少量的氟化物 some fluoride is added to toothpaste
... and other food-related expressions.