For those looking for how to do the ŵ in Tŵt Wales (Tŵt Cymru), it's unicode character u0174 for the capital Ŵ and u0175 for the lowercase ŵ.
On #Linux to type a unicode character press ctrl, shift and u, then type the relevant unicode character number, in this case it's 0174 or 0175. It takes a little effort to remember the numbers, but I feel it's well worth it.
@plantswithchris I'm not sure what kind of keyboard driver I'm running on my Manjaro, but it has a "compose key" which allows you to put together letters by typing multiple symbols in sequence. So for the letters you're describing, for example, I would press and release CTRL, then type a hat, ^, then type w, and it turns into the correct character. Most of the combinations are fairly intuitive. But this cheat sheet might be of help: https://fsymbols.com/keyboard/linux/compose/
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