The USA is one of the few, possibly the only, country which confers citizenship on people born there. Most countries confer the father's citizenship on the child. One of our sons was born in Switzerland, but there is no hope of his gaining that citizenship even if he wanted to.

Our son is British, his wife a US citizen and their children were born in the US. However, there is a time limit, long passed, during which they could have applied for UK citizenship for their kids. We didn't realize this until too late.

I would think it highly unlikely that you have any claim on UK citizenship as a consequence of being born there. You would also have a problem acquiring residency as a non-EU citizen.

The US is pretty tough on getting US citizenship: first you have to hold a Green Card for five years before you can apply and getting the Green Card is not easy unless you are sponsored by an employer. The citizenship application and test etc is pretty straightforward, but it may take over a year to complete the process. It is OK to hold dual citizenship for US and UK, but some countries, e.g. Germany require permission form their government to apply for US citizenship.

Probably more info than you needed, but it's raining here