If you are tax resident in the UK, you will pay UK tax on your workdwide income and gains with credit being given for tax paid on the same income in South Korea against your UK tax liability. See article 6 of the UK/South Korea tax treaty which deals with rental income here. You can assess your UK tax residency status here and here.
As your wife is not domiciled in the UK, she will have the choice whether to pay UK tax on her share of the non-UK income UK on the same basis as a UK national or to be taxed on non-UK income under the remittance basis rules which you can read about here (from page 54).
I hope this helps but let me know if you know if you have any further questions.
What if my wife is not domicilled in the UK (living but not resident) and the property is solely in her name, couldn't we get way with just paying tax in Korea and then spending it in the UK as just her savings? Plus, what if I hold onto my residency in Korea even if I spend some time overseas. Don't I just have to pay Korean taxes and not both?
What you need to do is to separate what money you have before you come to the UK from that you accumulate in Korea after you arrive in the UK. If you only bring money form your pre-UK savings account to the UK and can prove it if asked then you won't have to pay tax on it. If you bring the post UK arrival income to the UK, that will be taxed in the UK.You can be treated as resident in more than one country at the same time the separate rules of residency for each country. Article 4 of the tax treaty here does have a tiebreaker procedure to determine the overriding residency status of an individual. However, this clause cannot be used where an individual is only taxable in the UK or Korea on income from sources in that country. You cannot simply choose where you pay tax.