I have heard the America "Eagle" Russian "Bear" theory since I was in grade school (that's a long time ago). I've also heard arguments that the Antichrist will come out of Arabia (these are the least credible IMO). I have also seen good studies that place the AC out of the European Union. I personally lean towards the EU theory but am far from dogmatic about it. People far more knowledgeable than I argued out the credibility of each of these theories for many years with little gain towards any theory; and as it has nothing to do with salvation or my scriptural OBedience, I won't argue the point.
Having said that, here are some references I have had on hand that give some information on the word Tarshish. There is no support for the British Isles in either reference:
From Strong’s Hebrew Dictionary:
תַּרְשִׁישׁ Tarshiysh tar-sheesh'
prOBably the same as 8658 (as the region of the stone, or the reverse); Tarshish, a place on the Mediterranean, hence, the ephithet of a merchant vessel (as if for or from that port); also the name of a Persian and of an Israelite:—Tarshish, Tharshish.
תַּרְשִׁישׁ tarshiysh tar-sheesh'
prOBably of foreign derivation (compare 8659); a gem, perhaps the topaz:—beryl.
From the Jewish Encyclopedia:
Tarshish appears to have had a considerable trade in silver, iron, tin, and lead (Jer. x. 9; Ezek. xxvii. 12). It gave its name, besides, to a precious stone which has not yet been satisfactorily identified ( See Gems). The Targum of Jonathan renders the word "Tarshish" in the prophetical books by "sea," which rendering is followed by Saadia. Moreover, the term "ships of Tarshish" is rendered by Jewish scholars "sea-ships" (comp. LXX., Isa. ii. 16, πλοῖα θαλαρρης). Jerome, too, renders "Tarshish" by "sea" in many instances; and in his commentary on Isaiah (l.c.) he declares that he had been told by his Jewish teachers that the Hebrew word for "sea" was "tarshish." In Isa. xxiii. 1 the Septuagint, and in Ezek. xxvii. 12 both the Septuagint and the Vulgate, render "Tarshish" by "Carthage," apparently suggested by Jewish tradition. Indeed, the Targum of Jonathan renders "Tarshish" in I Kings xxii. 48 and Jer. x. 9 by "Afriḳi," that is, Carthage.
Josephus ("Ant." i. 6, § 1), apparently reading "Tarshush," identifies it with Tarsus in Cilicia. This identification was adopted by Bunsen and Sayce ("Expository Times," 1902, p. 179); but it seems from Assyrian inscriptions that the original Hebrew name of Tarsus was not "Tarshush." Bochart (in his "Phaleg"), followed by many later scholars, identifies Tarshish with Tartessus, mentioned by Herodotus and Strabo as a district of southern Spain; he thinks, moreover, that "Tartessus" is the Aramaic form of "Tarshish." On the other hand, Le Page Renouf ("Proc. Soc. Bibl. Arch." xvi. 104 et seq.) refutes this theory, declaring besides that Tartessus never really existed. Renouf's opinion is that "Tarshish" means a coast, and, as the word occurs frequently in connection with Tyre, the Phenician coast is to be understood. Cheyne (in "Orientalische Litteraturzeitung," iii. 151) thinks that "Tarshish" of Gen. x. 4, and "Tiras" of Gen. x. 2, are really two names of one nation derived from two different sources, and might indicate the Tyrsenians or Etruscans. Thus the name may denote Italy or the European coasts west of Greece.
Personally, where I see America in the Bible is in references like Isaiah 59:2 - 15
Isaiah 59:2 But your iniquities have separated between you and your God, and your sins have hid his face from you, that he will not hear.
Isaiah 59:3 For your hands are defiled with blood, and your fingers with iniquity; your lips have spoken lies, your tongue hath muttered perverseness. Isaiah 59:4 None calleth for justice, nor any pleadeth for truth: they trust in vanity, and speak lies; they conceive mischief, and bring forth iniquity. Isaiah 59:5 They hatch cockatrice' eggs, and weave the spider's web: he that eateth of their eggs dieth, and that which is crushed breaketh out into a viper. Isaiah 59:6 Their webs shall not become garments, neither shall they cover themselves with their works: their works are works of iniquity, and the act of violence is in their hands. Isaiah 59:7 Their feet run to evil, and they make haste to shed innocent blood: their thoughts are thoughts of iniquity; wasting and destruction are in their paths. Isaiah 59:8 The way of peace they know not; and there is no judgment in their goings: they have made them crooked paths: whosoever goeth therein shall not know peace. Isaiah 59:9 Therefore is judgment far from us, neither doth justice overtake us: we wait for light, but behold OBscurity; for brightness, but we walk in darkness. Isaiah 59:10 We grope for the wall like the blind, and we grope as if we had no eyes: we stumble at noonday as in the night; we are in desolate places as dead men. Isaiah 59:11 We roar all like bears, and mourn sore like doves: we look for judgment, but there is none; for salvation, but it is far off from us. Isaiah 59:12 For our transgressions are multiplied before thee, and our sins testify against us: for our transgressions are with us; and as for our iniquities, we know them; Isaiah 59:13 In transgressing and lying against the LORD, and departing away from our God, speaking oppression and revolt, conceiving and uttering from the heart words of falsehood. Isaiah 59:14 And judgment is turned away backward, and justice standeth afar off: for truth is fallen in the street, and equity cannot enter. Isaiah 59:15 Yea, truth faileth; and he that departeth from evil maketh himself a prey: and the LORD saw it, and it displeased him that there was no judgment.
Having said that, America certainly looks like an appealing place for the Anti-Christ. Perhaps he'll have a vacation home here...maybe in Hawaii