The USPTO is the United States Patent and Trademark Office (USPTO) is the federal agency for granting U.S. patents and registering trademarks. So no, it does not cover the UK. For the UK, you’ll search under https://www.gov.uk/search-for-trademark.
When considering which country to trademark your brand, you’ll prioritize the main market you wish to sell your product in. Therefore, considering the US is the largest market with the biggest sales opportunity we recommend you get your TM approved in the USA.
The challenges you faced with searching for ‘Majestic Peacock’ definitely sounds like a site error, so refresh your browser try running it again.
Amazon does have a new program that help expedite Trademarks — its called Amazon IP Accelerator. IP Accelerator connects businesses with a curated network of trusted IP law firms that provide high-quality trademark registration services to help brands secure a trademark, at competitive rates. This is a trusted intellectual property law firms at competitive rates.