I’m not sure that you really do get mutations in English, although ‘rooves’ might be the best example of the nearest thing. You do get assimilation of various sorts, as you do in most languages - where two sounds coming together cause one of them to change to be a bit less unlike the other. Quite often it’s the first sound that changes, as our mouths and brains get ready for what’s coming next: the ‘m’ of ‘comfort’ is actually made with the bottom lip and top teeth, like the following ‘f’, rather than with the two lips, like the ‘m’ of ‘mother’. (If you try to do a labiodental ‘m’ out of context you’ll just look like you’re doing a chipmunk impression.) Sometimes, though, it’s the second sound that changes - like the ‘s’ sounds of ‘cats’ and ‘dogs’, and it’s that kind of assimilation that really underlies Welsh mutations.
But then what happens is that languages change, and so the condition that leads to the assimilation can change or disappear. In Gothic the root form of the word that corresponds to Modern English ‘loaf’ was ‘hláib-’ (‘hl’ similar, but not identical to, Welsh ‘ll’, ‘ái’ like the word ‘eye’, ‘b’ like Spanish): it, or something like it, got borrowed into an early Slavonic language to give the ‘Polski chleb’ signs that you see outside Polish shops. When this ‘b’ was at the end of a word, or followed directly by an ‘s’, it became ‘f’ (Gothic hláifs, Old English hláf, Modern loaf); where it wound up between vowels, it stayed as a Spanish b/v (Gothic hláibos) or turned into a ‘v’ sound (spelt ‘f’ in Old English hláfas). In English this ‘f/v’ pattern has stuck, even though the following vowel has become silent (Modern loaves); and because it was quite a common pattern (thieves, leaves) it’s even got extended as a grammatical pattern to other words where the dictionary says we “shouldn’t”, such as ‘rooves’ (standard ‘roofs’).
So what happens in the Celtic languages basically follows this pattern, only much, much more so. A word like “yn” (=‘in’) obviously ends in a nasal sound, so the mutation in a phrase like ‘yng Nghymru’ isn’t really anything more than plain old assimilation. But then, presumably, a word like ‘fy’ must once upon a time have ended in a nasal, otherwise ‘fy ngardd’ and ‘fy nhad’ make no sense - so now we’re doing an assimilation with a sound that’s no longer there! In the same way ‘ei’ (=‘her’) and ‘ei’ (=‘his’) must have once ended in different sounds, or else it’d be hard to account for ‘ei phen’ and ‘ei ben’. But now we’ve got a situation where these changes, made in response to sounds that vanished from the language about the time of King Arthur, can in and of themselves convey meaning; and that then leads us on to changes like ‘chlywais i ddim’ (‘I didn’t hear’, from ‘clywais’ ‘I heard’), where the word ‘nid’ (‘not’) - which has vanished - used to have some sound on the end - which has also vanished - that caused a following ‘c’ to turn into a ‘ch’. That change in its own is now fixed and meaningful, such that before you even get as far as ‘ddim’ we know that you haven’t heard, even though all the original, underlying, historical reasons for that change basically never made it back from Catraeth
So, yeah… we can sort of find phenomena in English (and other languages) that kind of parallel mutations, and ‘rooves’ is probably the best example but, frankly, it ain’t a patch on Welsh