We even have place names in Welsh different to the English for places accross the border (Llundain for London), often reflecting the fact that the UK was Celtic before the Romans, Saxons and Normans came to play.
I think it'd be more correct to put that into the "Welshification" category, you mention next. 'Londinium'
has been suggested as having Celtic origins, but many other sources (including pre-Celtic languages) are as likely, and the difference of the Welsh name from the current English name is so little I think there's a good argument that the divergence (one way or another) is fairly recent, making it likely to be English->Welsh in nature.
Certainly, for a significant amount of post-Roman history and probably also post-Saxon history (with various tensions in the landmass marking significant periods before "England" came to exist, as opposed to the multiple kindgdoms, rebooting the practice quite a bit until it may have stablised somewhat into the last thousand years or so), Welsh drovers would drive their livestock to places (including London) outside of Wales, and coming back again. Saying something like "I've been to London" in Welsh would have probably have caused the something like the "Llundain" name to be known and understood within Wales, purely verbally, sometime prior to the first successful/popular attempt to set down the spoken language in writing. At some point after literacy established a convention (whether amongst the masses, or just amongst the earliest chroniclers) it would have solidified into something like the form we see now. Liverpool is certainly another (closer!) destination that would have had the same word-of-mouth transportation of its name into the Welsh heartland.
Something makes me want to imagine, BTW, that "Llundain" is something a little bit "Cymglish" in nature, making it a polar reverse of the Abermaw=>Barmouth mutation, the visiting Welsh (mis)translating in their own way.
After 'travel writings' and knowledge of the wider world gained hold, names would largely no longer be distorted, so. You can probably derive some interesting timeline-maps of what places were popularly known in what foreign locales by mapping the discrepancies with the local names. As far as I know, New York is known as "New York", more or less worldwide, it having become so (after abandoning New Amsterdam or other variants beforehand). Munich/Munchen is a mild example, and there's also the Milano/Milan and Roma/Rome examples of 'mild' Anglicisation. Lesser-known places that got known only later (or much well more known ones that had already solidified in scholarly writings) are more likely to be known in English by their local names. (If not
pronounced exactly the same... c.f. "Paris".
(What I find interesting is how Deutschland/Germany/Allamange/etc is known, in each near-neighbour
foreign language by the specific group of people that were known by the peoples that contributed their description to the modern-day. Through the Romans, we and a number of other countries have a variant on "Germany", other countries name from the Alamanni tribe, or the Saxons, the Nemich(sp?) or, or course there's the local word that became "Deutsch"-ish.) More far-afield nations will have been influenced by colonial/diplomatic exporting of a specific country's terminology. I think Japan is rooted in Germany's own name, While Hawaii got inspired by the English. Back to the Welsh, it's the Francophone name that came to influence them.)