Method 1 (recommended): To avoid going in a big circle in the gov.uk website:

• If you have lost your User ID, it should have been emailed to you when you registered online, no doubt, so check your email account for an email subject 'Your Government Gateway ID'.
• If you have deleted or cannot access your email you will need to phone HMRC for help. It probably means that you need to register though.
• If you have found your ID go to www.gov.uk//hmrconline and enter your ID, hopefully you will have saved your password on your computer and can now login. If not, phone HMRC.

Method 2 (not recommended): To go in a big circle, follow the instructions on HMRC:

Don't bother trying to find your ID online as it will take you in the following circle:

Step 1: I went to www.gov.uk/hmrconline to activate my new business.

That took me to https://www.gov.uk/log-in-register-hmrc-online-services

Step 2 I found that I had lost my ID and password so I clicked 'Get help if you’re having problems signing in to your account.' to follow this link www.gov.uk/log-in-register-hmrc-online-services/problems-signing-in

Step 3. I selected 'recover your user ID and password' 

to follow this link https://www.access.service.gov.uk/account-recovery/forgot-userid-password/check-your-emails?_ga=2.109131970.637892711.1692357858-285498506.1683123296

Step 4 took me to a page which said...

You cannot access the service from this page

To access this service, you must return to the government page you came from and start again.

You can also start again from the GOV.UK home page.

You might need your Government Gateway user ID and password.

Make sure your web browser can accept cookies.

Go to GOV.UK home page

Step 5. I went to Go to GOV.UK home page and typed in 'help with user ID and password' and it took me back to 

https://www.gov.uk/log-in-register-hmrc-online-services

Step 6. I phoned the helpline, I said that I would complete the survey so that I had more chance of the phone being answered. It took 15 mins but eventually spoke to a nice chap who said:

'Check your email for your email'.

The email I found which I had thought was lost (we just changed our servers) is called 'Your Government Gateway ID'.