1) U.S. citizen is taxed on worldwide income.
2) In your first paragraph, the primary residence cost GBP 300,000 in 2010.
In your third paragraph, this residence cost GBP 510,000.
3) I think you meant that the house is sold higher than U.S. $250,000 home owner exemption exclusion. Please double check on both U.K. and U.S. law to make sure that the gain on the house sold is reported properly. This is the portion you want me to assume with no problem.
4) It seems that your question is on whether we need to report earnings on foreign exchange rate.
Foreign exchange rate dealing is a common way to make money and invest.
Yes, earning on foreign money trade and exchange needs to be recognized as income.
The mortgage was taken out in British Pound and paid in British Pound. There is no exchange rate difference.
5) In the taxpayer's situation, the only number in question is the 200,000 in 2010. But she was not a corporation engaging in the investment activity. The exchange rate happened in between 2010 and 2016 for six years should not be an investment issue. That is, if she is on the other side loosing big for that type of currency exchange, she cannot claim loss. Also, if she stays in the same country, there is no issue.
At the moment when she move money to another country, she may experience increase. But if she stays, she should realizes the difference of living standards, etc., those money will be gone soon.
In her situation, there is no foreign exchange gain issue. If she moves the money back to U.K. the next day, she will lose all these phantom gain. However, if she actively trades exchange rates, she will be liable for the proper reporting of the gain of such trades.
6) Be award of this Sterling asset she is investing. It may not be sound, and bona fide investment. Double check.
Please feel free to follow up.