Assuming a time where appointing a Jewish king is possible, if a proper Jewish king was anointed would he continue to retain the status of king if the Jewish people were exiled from the land of Israel?
Rav Aharon Lichtenstein discusses this sort of issue in an article of his titled "סמיכה בארץ ישראל ובחוץ לארץ" (roughly: ordination in Israel and the Diaspora) printed in his collection of essays Minchat Aviv beginning on page 479 (particularly pages 481-483).
Rav Lichtenstein opens his discussion by citing the Tosefta (Sanhedrin 4:6): "אין מעמידין מלך ישראל בחו"ל" -- We do not establish kings in the Diaspora.
He notes that while the Rambam in his Laws of Kings mentions a geographic limitation in his listing in Chapter 4 of the king's powers (namely, in 4:2 ושולח בכל גבול ישראל (cf. Hilchot Shemitta 10:10 and Vayikra 25:9)), he omits such a mention in Chapter 2 where he lists the obligations related to honoring the king (in particular, in 2:3 where he doesn't suggest the widow perform Chalitza in the Diaspora).
Accordingly, Rav Lichtenstein argues that a king ruling in Israel has full power as king only in Israel, while the obligation to honor him extends elsewhere.
However, a pseudo-king ruling in the Diaspora is different. Rav Lichtenstein argues that since there is no separate Mitzva of honoring a king as there is by Talmidei Chachamim (Chinukh #222) [or parents (Chinukh #27)] but rather it is a part of the nature and Mitzva of appointing a king as opposed to any other leader, a pseudo-king ruling in the Diaspora would not even need to be honored there as his formal powers do not apply, as above.
Since, as he notes in footnote 6, the Yerushalmi (Horayot 3:2) seems clear that kings can gain and lose their status depending on de facto leadership (eg. through a temporary coup, such as with David and Avshalom), it seems to me that, combining all the above, the individual in your question would no longer have the status of a king, neither regarding his powers nor the obligation to honor him, since he no longer has any de facto control of any territory in the Land of Israel. (It may still be appropriate to honor him if he is the de facto ruler in his current location (or perhaps just as a remembrance of the past), but it is not part of the formal biblical rule of kingship so much as part of Dina deMalchuta.)
[I don't see anywhere where he deals with the fact that some laws (eg. those in 2:2) of honoring a king still apply after the king dies and is (presumably therefore) no longer king. Perhaps we should distinguish between a king who dies in office and one who leaves before dying, or perhaps between those laws honoring the person of the king versus those that honor his office. These would of course also then be relevant in understanding II Samuel 16:20-23 and 20:3.]
Note that Zerubavel was not treated as a king even though he would have become the king (as the desxcendant of malchus bais David) had they been completely redeemed. Yehoyachin was still regarded as the king (at least in terms of continuing the Davidic line). I point out the descent through Zerubavel in which kings must Moshiach descend from?
Thus we see that Yehoyachin was still considered a king and Zerubavel and his descendants were potential kings.
All descendants of the house of David have Kingship as an inheritance. This is the plain meaning from the "Samcheinu" prayer said following the reading of the haftorah each Shabbat.
"We rejoice HaShem, our G-d, in regard to Eliyahu the prophet and in regard to the Kingship of the house of David, Your anointed. He will come speedily and our hearts will rejoice about his throne upon which no stranger sits. And no others will inherit his glory. Because with Your holy name, You swore to him that his candle will never extinguish ever. Blessed are You, HaShem, shield of David.
This inheritance by the descendants of the Davidic line is in all places and all times.
Anointing is something that is done for non-Davidic line kings or when there is more than one potential king making claim in the Davidic line. The anointing, which was generally done by a prophet, resolves any question as to who becomes King.
What your question seems to be more directed toward is whether someone has the "din of Malchut" during exile. Meaning, are the halachic obligations of the King's subjects present during exile? And if so, are the legal consequences for failure to meet those obligations also present?
According to Sefer Yechi HaMelech, based upon the teachings from the Lubavitcher Rebbe, this is one of the primary distinctions between the status of "Presumed to be Moshiach" and "Moshiach with certainty". The final halachic status of "Moshiach with certainty" carries with it the obligations of the subjects toward the King and the consequences of failure to perform those obligations.
Another consideration with this question is in regard to when the nation is required to appoint the King. This is discussed by the Tzemach Tzedek in Sefer Derech Mitzvotecha and also by HaRav Yitzchok Ginzburg in Sefer Malchut Yisroel. Although this is associated with the return of the Jewish nation into the land of Israel as the ruling government, it is generally held that thcommandment becomes obligatory when the majority of the global Jewish population resides within the borders of the land of Israel. This is one of the compelling ideas of the current time. Because each year, the census in Israel shows we are extremely close to that tipping point, if it hasn't actually been achieved already.
The general laws for both Kings from the house of David and those of other tribes can be found in Mishnah Torah, Laws of Kings and their Wars, chapter 1, laws 7-9.
7: "Once a king is anointed, he and his descendents are granted the monarchy until eternity, for the monarchy is passed down by inheritance, as Deuteronomy 17:20 states 'Thus, he the king and his descendents will prolong their reign in the midst of Israel.'
If the king leaves only a young son, the monarchy should be held for him until he matures, as Yehoyada did for Yoash. The order of inheritance of the monarchy is the same as that governing the inheritance of property. An older son is given precedence over a younger one.
Not only the monarchy, but all other positions of authority and appointments in Israel, are transferred to one's children and grandchildren as inheritances forever.
The above applies if the knowledge and the fear of G-d of the son is equivalent to that of his ancestors. If his fear of G-d is equivalent to theirs but not his knowledge, he should be granted his father's position and given instruction. However, under no circumstance should a person who lacks the fear of G-d be appointed to any position in Israel, even though he possesses much knowledge."
8: "If a prophet appoints a king from any other tribe of Israel and that king follows the path of Torah and mitzvot and fights the wars of God, he is considered as a king, and all the commandments associated with the monarchy apply to him.
Although the kingship was primarily given to David and one of his descendents will be serving as king, there is halachic legitimacy to the rule of other kings. Behold, Achiyah of Shilo appointed Jeroboam and told him (I Kings 11:38): 'And it shall be that if you obey all that I command you... I will build you a faithful house as I built for David.' Similarly, Achiyah told him (ibid.: 36): 'To his (David's) son, I will grant one tribe, so that David, My servant, will always have sovereignty before Me in Jerusalem.'"
9: "The kings of the Davidic dynasty will prevail forever (II Samuel 7:16): 'Your throne shall be established forever.' In contrast, should a king arise from other Israelites, the monarchy will eventually cease from his descendents. For behold, Jeroboam was told: I Kings 11:39 'I will afflict the House of David.... but not forever.'"