Having a word trademark already accepted in the US can potentially facilitate the process of trademark registration in other jurisdictions, but it does not guarantee automatic acceptance or make the process universally easier.
A US trademark registration can be advantageous, as it serves as evidence of your legitimate use and ownership of the trademark. When applying for registration in other countries, the fact that you have successfully obtained trademark protection in the US can demonstrate the validity and strength of your mark, potentially strengthening your position during the examination process.
However, it's important to note that each country has its own specific requirements, laws, and examination processes for trademark registration. While a US trademark registration can provide certain advantages, it does not guarantee automatic acceptance or exemption from meeting the specific criteria of each jurisdiction. You will still need to comply with the relevant laws, regulations, and procedures of the countries where you seek trademark protection.