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If you are resident in the UK and receive rental income from an overseas property, you will be taxed on this in the UK in the same way as if the property was located in the UK under the income tax rules. The first £1000 of your income from a rental property may be tax free because of the UK's property allowance.
If a double taxation agreement gives exclusive taxing rights to the UK, then tax is only paid in the UK (and vice versa). This overrides the domestic UK law and the domestic law in the other country. So do not report as income.
If there is no double taxation agreement in place in the jurisdiction where your overseas property is, then you may still be able to get tax relief for any foreign taxes you pay against your UK tax bill through a foreign tax credit (or unilateral relief).
The foreign tax credit is the lower of (a) the foreign tax paid on the income or capital gain and (b) the UK tax liability on that same income or capital gain.
That is correct. If only the other country is allowed do not report that on your UK form
Yes you should do that