They're both unusual name choices as they don't seem to be names necessarily, but descriptions of what they are.
This leads me to believe the UK inspired the US's name, considering their interwoven history.
No, it did not -- while the Declaration of Independence (1776) is the "unanimous declaration of the thirteen United States of America," the term "United Kingdom" was not used for the political union of the kingdoms of Great Britain and Ireland until after the Acts of Union (1800) actually united them. The OED has a reference to a "united Kingdom" from 1690 and another from 1737, but it wasn't until the 1800 Acts that the formal name came into being. The prior "united Kingdom" was a personal union.
The OED references for context:
1690 tr. G. Petyt Lex Parliamentaria (ed. 2) xi. 217 A like Clause for incapacitating Persons to be elected, &c. Members of Parliament, and likewise for incapacitating Members of Parliament, with like Restrictions, Exceptions and Penalties (throughout the united Kingdom) as are contained in the Statute 4 & 5 Annæ.
1737 Gentleman's Mag. Oct. 609/1 I have more Reason to oppose it, than any Man in this House, nay perhaps than any Man in the United Kingdom.
1800 Act 39 & 40 Geo. III c. 67. 359 The said Kingdoms of Great Britain and Ireland shall..be united into one Kingdom, by the name of The United Kingdom of Great Britain and Ireland.