This is the first variant:
The reason there are two variants of each name is the peculiarities of the Mongolian spelling: in the Classical Mongolian alphabet there are some letters which are written one way when they appear in Mongolian words, and in a different way when they appear in a foreign, borrowed word. Both names are not Mongolian, so they are borrowed.
Letters U and K behave like this, so I made 2 variants for "Lucas": a) is the way the name could be written if it were a Mongolian word, b) is the way the name must be written because it's not a Mongolian word, the U and K letters are the ones used in foreign words.
"William" also has two variants but the reason is a bit different: Mongolian words never have the LL letter combination, and the very shape of this letter this letter is opposing the doubling.
So I made: a) the phonetic rendering of the name "William", a Mongolian person would write it this way, because there's no phonetic difference between English LL and L, so it's just WILIAM, and
b) the word transliterated completely, with double L.
If you like it, that's great. May be there are other people here who can comment on which variant is "more Mongolian", or correct my mistakes, if any.
P.S. Here's a rare Mongolian book: a manual of the Mongolian script for Mongols who used only Cyrillic alphabet. It's great in one aspect: there are many transliterated Mongolian texts.
I scanned it:http://www.uz-translations.net/?category=mongol-mongolbooks&altname=monggol_bichig_-_mongolian_script
Register there to see the links to download.