Will my employer see my health check results?

I'm going for my first health check next week but there are some aspects that will come out that I really don't want my employers to know about.

What kind of information do they get back from the clinic? Do they get told everything that happens? Do they just get told that I've been?

They have all the info about your health issue, since the first year you have been working there. That's why they ask you to do health check. This is compulsory by the employment law.

I used to have a health check every year with my old company, and the results were 100% private.
The health check at the company I work at now is optional, and results are still 100% private.

Perhaps there may be some exceptions, but I can't see a clinic providing the results to the company.

Depends on company, some have the checkup result send to company address while some address directly to you.
I experienced both and from my experience both ways the result is only for you to see. It's a private matter. Although one employer did request for a copy as records.
So check with your colleague how they go about they health checkup results.