Liam has long been a problem name for the heralds. Many people assume it is medieval, but in fact we currently have no evidence of its being used as a given name in the medieval or Renaissance era. Nor has there been any evidence of its being a period nickname for "William." The closest thing to Liam we have found so far in an Irish context is a woman named Joan nyne Lyeme (in other words, Joan daughter of Lyeme) in an Anglicized Irish record from the era of Elizabeth I.
However, for people determined to have the name Liam (I'm looking at you, +Bill Toscano ), we have found a way to make it registerable. As I have discussed a few times in these articles, there is an established a pattern of using late 16th century English surnames as given names. The most well-known example of this is Guildford Dudley, the husband of Lady Jane Grey, whose first name was based on a family surname.
We do have evidence of a woman with the surname Liam who was married in Cambridge, England in 1592. Because the pattern of using 16th century English surnames as given names actually existed in period, evidence of Liam as an English surname permits registration of Liam as an English given name. As it is considered an English name, it can be used with an Irish or Scottish surname.
 The Fourteenth Report of the Deputy Keeper of the Public Records
in Ireland (https://archive.org/details/reportofdeputyke1416irel),
 September 2012 Cover Letter
 Joana Liam; Female; Marriage; 1592; Elsworth, Cambridge, England;
Batch: M13053-1 (https://familysearch.org/pal:/MM9.1.1/NJ1W-6MF).