Only if the other country demands it. The US allows dual citizenship.
Tax implications only really affect you at high income levels (think millions of USD/annually). The cases where you are considered 'double taxed' are, IMHO, rare, and in fact mostly do not happen. This comes from a person who detests taxes of all kinds.
There are guides out there including by the US government detailing how expatriates can manage their taxes, suggest reading on them.
The cost of getting a US / UK accountant to do the most simple of returns will run you around $1000 in total (at least in the UK), and the costs increase dramatically as your taxes get complicated.
That doesn't even get into things which aren't covered by the DTA, like ISA accounts in the UK. And worse it's extremely easy to run afoul of PFIC rules which lead to punitive US tax and a higher reporting burden if you do any investing.
Being a US expat is expensive if you don't want to give up your citizenship.
>Tax implications only really affect you at high income levels (think millions of USD/annually).
Depends on what you mean by "really affect" because they definitely affect you at any income level as you have to file taxes with the IRS.