I note:-
(Beurla) This is the song of the sword of Alan
(Oidhirp) Seo an t-òran a' chlaidheamh Ailean
(Nas fheàrr) Seo òran chlaidheamh Ailein
This shows a lot of things. Firstly, the RLS English "tronslotion" (I doubt it actually was...) is somewhat "Gaelicised" in terms of word order etc., inasmuch as a native English speaker would more likely say something like "This is the song of Alan's sword" (although probably not "This is Alan's sword's song"...).
Secondly, English has a somewhat strange "genitive" structure - as illustrated in ultimate form by the final rendition in parentheses. That is, the "genitives" come first, all with their "genitive markers" (usually the letter S with an appropriately-placed apostrophe) and all articles - definite or indefinite - (words like "the" or "a") - vanish apart from the first one. However, this rather odd genitive structure can be avoided by using - as is done in the RLS "tronslotion" - the word "of" before the noun to indicate genitive. However, the result can look a bit stilted, as in "the song of the sword of Alan"
Aside: using a mixture of English genitive forms - as suggested in the "natural" sounding "the song of Alan's sword" - can be ambiguous - especially when the first two items often form a "stock expression", e.g. "the chief of staff". For an expression like "the chief of staff's concerns" could be interpreted as "the chief concern of staff" or as "the concerns of the chief of staff": the S-genitive in English can "float" to make a "stock expression" like that genitive.
If one examines the Gaelic I have labelled "Nas fheàrr" (better), you will notice a complete lack of articles. This is because the final item is a proper name and hasn't got an article - and because all articles before the last one vanish in such a "genitive chain". So, just as in English it would be wrong to write, e.g., "the sea's the house", so it would be wrong in Gaelic to write "an taigh na mara". In Gaelic - just as in English - it's only the actual "genitive" thing "na mara" - "the sea's") that keeps the article - the other bit loses it (just "house" - just "taigh").
Another thing to note is that Gaelic still has a "case" system - as has English. In both languages, the "case" system has become somewhat rudimentary, but it hasn't completely vanished - as it has done in e.g. Welsh. In English, "genitives" are almost always shown by "'s" (singular) or "s'" (plural), with pronouns providing the only exception I can currently think of. In Gaelic, it's not quite so straightforward: some nouns just don't change at all in the singular genitive (e.g. "cumhachd" - power), while others get thinner (as in example - e.g. "Ailean" => "Ailein") and a few add an A (often with other changes, e.g. "druim" => "droma" - rigg). Short female-gender nouns often add an E, e.g. "sgoil" - "sgoile". Unlike English, however, such changes (a) only affect the LAST in a "genitive chain" and (b) are also seen in ADJECTIVES.
Anyway, "the song of the sword" would be "òran a' chlaidheimh". However, if you add Alan into the picture, the article before "chlaidheimh" vanishes - and "chlaidheimh" reverts to its "standard" form (claidheamh). Personally, that's the form I would use in "the song of Alan's sword"/"the song of the sword of Alan", though the "better" tr*nsl*t**n quoted has a sèimheachadh (lenition) on "claidheamh" still: I'd only do that if "òran" was female gender, which it isn't.
So, putting it all together:-
* the song - an t-òran
* the song of the sword - òran a' chlaidheimh
* the song of the sword of Alan - òran claidheamh Ailein
Genitive plurals in Gaelic - if you need them - if they have the definite article, it's just the genitive plural with "nan" in front (genitive plural form of "the"). Without an article, it's just the genitive plural - but with sèimheachadh if it can be shown, e.g. "a Pict" - "Cruithneach"; "the Pict" - "an Cruithneach" - or "a' Chruithneach" if female; "of a Pict" - "Cruithnich"; "of the Pict" - "a' Chruithnich" - or "na Cruithnich[e]" if female; "Picts" - "Cruithnich"; "of Picts" - "Chruithneach"; "of the Picts" - "nan Cruithneach"
Note how this noun - and in general nouns that make their plurals with thinning (by changing the last vowel, as in "foot" - "feet" in English) - revert to the "singular" form for the genitive plural. Nouns that DON'T do this usually keep the ordinary plural form for the genitive plural.
Duilich - sorry - rabbited on too long...