How difficult is it to find a job in the UK as a newly graduated HSE engineer with an MSc in HSE?

Finding a job in the UK is not different from anywhere else. You need to apply and apply, over and over.

In terms of qualification, you should be fine with MSc in HSE, nevertheless NEBOSH and UK National Vocational Qualification (NVQ) are preferred to get a job in the UK, because the employer may not be an expert in HSE and not know how his business can benefit from each piece of cert you possess, therefore having something more common to the country should be something to look into.

And you can always find some free training available on EntirelySafe.com ( http://entirelysafe.com/all-courses/ ) that can help you to get more credit while seeking employment.