NEBOSH General certificate will cover just about everything in working life. The construction certificate will cover the more exciting bits of construction equipment and the different mindset that exists in the industry.
If you can do some environmental stuff and quality along with it you will be well set.
While the number of HSE inspectors will reduce the insurance industry employs lots of H&S qualified people as do consultancies and within large companies there are numerous qualified people. Insurance drives the H&S culture as they look to keep costs down on the claims front.
My experience was that construction is a world apart where they will regard you as the enemy. You will be lied to and teams will conspire against you to avoid you marking them down, affecting their pay. Main contractors will expect you to do your job and stay out of their hair. There’s a lot of slopey shoulders where no-one wants to be the company paying for the safety.
Not all consultancies are much cop some are a box ticking exercise to meet the requirements for the main contractor/insurer. Try to find one that expects you to spend the time required to do the task and will also support your Continuing Professional Development with time and or finance.
In the current climate you will need to get on with the diploma as soon as you have your general certificate.