This is FAR from my area of expertise, so could be completely and wholly incorrect.
But this is what I would anticipate....
UK taxes are typically higher than US taxes. When any single year is looked at, I would expect that you will have paid tax to the USA for the rental income. And then additional tax to the UK on the balance. I do not believe the personal allowance exemption of £12k ish is available for property income. This is one of the many many barriers HMRC have put in place to make being a landlord undesirable.
For fun, let's say you have $10,000 income per year (I like round numbers).
File US tax and if the tax rate on rental income is 15%, you'll pay the IRS $1,500 in tax.
File UK tax and if the tax rate on rental income is 20% (and I suspect there is a treaty preventing double taxation on rental income). So the full UK tax would be $2,000. As you've paid the IRS $1,500, you'd only by HMRC $500, the balance to the UK tax amount.
If you were to pay HMRC first, you'd pay $2,000. Then you would input the foreign tax of $2,000 pain on your IRS return and would not owe the IRS on that as you've paid tax under a treaty between the USA and UK. And because the UK tax is higher. That will always be the downfall. The country with the higher tax rate will take that "extra bit". It's not double taxed but it is taxed in full, if that makes sense. You'll lose the benefit of "just" paying the lower tax rate.
There will be a lot of moving parts to this and I'd recommend reaching out to Monument Tax who are a reputable, affordable US/UK tax firm.
Moving on from that, I can't help but think there is likely tension and possibly a disagreement with your sister about the home. Needless to say if she's been running it as an AirBnB for 3 years and you would like to sell (guessing?), I'm sure that has strained the relationship. And it would be a shame as the housing market is just settling a bit after going CRAZY for a good 6 months.
I hope you are able to make headway in a resolution when you see her next. I'd recommend having a few valuations completed on the property and asking her to just buy you out). If she doesn't have cash, she could take a mortgage out for half the value that she needs to pay you. She will need to get over this.
Has probate finished and has the estate been settled?