If I were to customise the glyphs of Gotham Rounded/Linotte Heavy to suit my requirements, what would the implications be with regards to copyright? Surely, if I were to do the same and it had visible differences on some glyphs, I could create a font influenced by the above two fonts? I'd be very interested to hear your thoughts on the issue.
First of all, this might be outside the scope of the question but Linotte and Gotham Rounded are both low-contrast sans with rounded terminals, but they're actually quite different. It's especially obvious in characters like M and a.
Now, to your quesion about creating a rounded version of an existing commercial font:
First of all, you would have to check the licence which came with the font. Most licences don't allow you to do any modifications to the font. Specifically Hoefler (who licences Gotham) says this about modifying:
Once converted to outlines in a drawing program, you can alter the shapes of letterforms to produce a customized piece of typography, such as a logo. But you can’t alter the fonts themselves, or use them to make new fonts, nor engage anyone to do this for you. Such adjustments are considered “derivative works” under United States law, and they require the prior written permission of Hoefler & Co. as their copyright holder. You’ll find this described in greater detail in your End-User License Agreement.
Like it says, not only is it against the licence, it would be considered a derivative work, unless you start off drawing every glyph on your own and then be only "inspired" by other existing fonts.