I may be wrong about this, so don't take my word as legal advice, but as far as I know there is no law or regulation that forces you to disclose your autism diagnosis to an employer. I gather that it's completely at your own discretion whether or not to choose to tell a boss, co-worker, etc.
Some people never disclose. Others disclose to an employer or for university/college in order to be granted certain reasonable accomodations that would help them perform better at work, for example if you have to explain that the reason you need a quieter office space, or to be away from harsh lighting or noise, is because your autism makes you less able to cope with those conditions. Then I believe your employer is legally bound to make attempts to meet those needs as long as they are "reasonable accomodations." But you are under no legal obligation to reveal your diagnosis unless you need to obtain this kind of help in the workplace, or you need benefit assistance etc.
This is as far as I'm aware, anyway.
If you obtain a private diagnosis instead of NHS, the privacy of your diagnosis is even more under your control, although again I'm not sure but if you ever need benefits or other more formal, legal assistance due to your autism, I gather only the NHS diagnosis is accepted as qualifying you, but I'm not 100% sure.