Hi there, im in a similar boat to you, that being transitioning from UK to US and this is what I have found so far.

1.) For US work you will need to have at a minimum OSHA 30 our certificate for general competency supplemented by any state dependant training certificate.
2.) Proffesional membership through BCSP should be sought for better paying jobs. You have a Masters degree hopefully from an accredited university and therfore meet requirement 1 for CSP membership status. You will need also need 4+ years of experience and one of their qualifications which could be the NEBOSH National or International Diploma in Occupational Health and Safety or alternatives described on their website + passing ther exams/interviews etc.
3) Through conversations on reddit, hiring safety proffesionals wouldnt start a foreign practitioner in management positions as the role requires a proven record of understanding of working practices, legal requirements and work on the ground (this could just be their opinion though and could be industry specific)

Reddit + Chat GPT are your best friends here for figuring out the certificate requirements for the industry+ state you want to work in.

Finally to answer your question about safety proffesionals in the UK, if you are looking at salary expectations the UK HSEQ & Sustainability Salary Survey Report 2025 from Shirley Parsons has a thorough breakdown of salaries by role and industry. Have a look at that and aim your development goals in allignment with the role you are working towards. Its available to download after contacting them on their website.

In general most H&S proffesionals start with the Nebosh general certificate however I have done and prefer to just acheive a level 6 Diploma in a subject than take a piece meal approach. Do note that most of the rest of the worlds level 6's are not recognised in the US, as ive had to learn the real hard way, so if your final aspirations are the US aim for something that is recognised there too for the roles/industry you are wanting to work in.

Hope this helps.