Besides the fact that Londres and so on originate from Latin Londinium, I unfortunately have not been able to find any dictionary entry that explains the etymology of this word and the sound changes that occurred.
However, I did find a post on the Wordorigins Discussion Forum archive that gives some information:
It's a regular sound change in Old French whereby Cn > Cr in
post-tonic (after the stress) syllables. Londiniu(m) > *Londne >
Londre (the Old French form), just as ordine(m) > *ordne > ordre
'order' and diaconu(m) > *diacne > diacre 'deacon.' – language hat
There seem to be similar examples of this sound change in Spanish. Using the Diccionario de la lengua española online, I found the following two examples of a Vulgar Latin sequence -ndin- becoming modern Spanish -ndr-:
liendre < lendinem (V. Latin *lendis, *lendinis)
landre < glandinem (V. Latin *glando, *glandinis)
These words may not be exactly analogous to Londinium, because they have a short i. I haven’t been able to find a definitive statement on whether the i in Londinium was short or long; Wikipedia suggests that it is unclear.
But it looks like the phonetic result was the same: in any case, the first i in “Londinium” was elided, and the resulting cluster /ndn/ was changed to the easier-to-pronounce /ndr/.
The sound /l/ could also become /r/ in this context, as seen in Spanish alondra from Latin alaudulam. However, it doesn't seem necessary to postulate an intermediate change of /n/ to /l/ to explain the words above with -ndr- from -ndin-.
I don't know if the change of n to r in these contexts is a common sound change from before the Romance languages diverged, or if it was a parallel development, possibly due to regional influence between the languages that show this change.
In Spanish, n furthermore became r after other nasals in some cases, with an epenthetic voiced plosive inserted in between, as in hembra from Latin feminam or hombre from Latin hominem. However, it seems nasal + nasal clusters developed differently in French: the cognate French words femme and homme don't have r. (Actually, the situation in Spanish seems to be more complicated than I initially thought, since there are also examples like dueño < dominus; see the following WordReference Forums threads for more discussion: "Spanish words of type homBRe & hemBRa" and "Londra, Londres - the origin of the /r/ sound in Romance for "London"").
I don't know how the stress ended up on the first syllable ("Londinium" would be stressed on the second syllable according to the regular Classical Latin stress rule, but the Latin stress system evolved, partly due to things like loss of distinctive vowel length, so stress sometimes falls on a different syllable in the Romance languages vs. in Classical Latin).