Should you tell HR about neurodiverse conditions?

If you will need any change or accommodation in how your are treated or trained, and or in expectations on your performance, you need to speak with your supervisor. They can help decide if it is something that needs to be brought to HR or not. Neurodivers conditions like Autism and ADHD fall under the ADA, so an employer with more than 50 employees is required to make a good faith effort to accommodate your needs. 

The only reason anyone would tell other employees about your condition is if it is necessary. For example, if your boss wants another employee to walk you through a process and you'll need that done for you several times, it would be valuable for the other employee to know how to handle that situation.When you are interviewed, if you feel anything about the job will  be an issue for you, it is your responsibility to bring this up.  Sometimes employers will ask "is there any reason you won't be able to perform the duties as described?"

No need to, I don't have one.

If you need certain accommodations to be able to perform your work successfully, you will need to tell HR once you've been offered the position. There's nothing wrong with asking that your needs not be discussed with anyone other than your direct supervisor.  It may become readily apparent to the rest of the staff though without anyone telling them.  Be very careful of using it as an excuse for anything that might go wrong. They have to make any reasonable accommodations for you, but you have a responsibility to do the things that you need to do in order to perform your job duties.  If you need to write down instructions or notes and refer to them often, then do so.  Saying "I couldn't get it done because Sam was breathing too loud" isn't going to be acceptable.

Since you haven't started, you must also be a mindreader to know that you will struggle to perceive information and instructions.

If you want to be treated differently, you need to request reasonable accommodation through HR.  This will include specifically what you need from the company to succeed.  HR will evaluate the information and determine if it's feasible and who needs to be aware of the situation.  If your peers are training you or working hand-in-hand with you, the company can determine that they need to know.

Be mindful, if the company is looking for an employee that has specific traits and that's not "you". The company wouldn't keep you employed simply because you are neurodiverse. You have to be able to do the job you are hired for / assigned to.

I'm not in your shoes.  We don't know what you mean by neurodiverse as it's an abstract phrase.... without knowing the details, it seems the only special treatment you can request at this point, is weekly one on one sessions with your manager to confirm you are grasping the job and if not, determine changes that are needed to help you succeed. 

HR is not supposed to let anyone know of any of your personal and private things...medical status being one of them.  If it is relevant, then your line manager will have to know.  If any other employees ever pop up and 'know' your medical conditions...well...in that case you can sue.