I know that WebKit relies on either libsoup [1] (on Linux/Unices) or curl [2] (legacy Windows and maybe WPE(?)) as a network adapter, so the header handling and parsing mechanisms would have to be implemented in there.
Though, on MacOS, WebKit tries to migrate most APIs to the Core Foundation Framework, which makes it kind of impossible to implement as a non-Apple-employee because it's basically a dump-it-and-never-care Open Source approach. [3]
Don't know about chromium (my knowledge is ~2012ish about their architecture, and pre-Blink).
I wasn't sure about WPE in regards to libsoup due to the glib dependencies and all the InjectedBundle hacks that I thought they wanted to avoid.
I mean, in principal curl would run on the other platforms, too...but as far as I can tell there's an initiative to move as much as possible to the CF framework (strings, memory allocation, https and tls, sockets etc) and away from the cross-platform implementations.
