Hello @BasRieter!
Welcome to the community forum!
You can have a look at the following thread that also talks about the subject:
In short you have to create two SonarCloud organizations, one for Github while connected with your Github account and one for Bitbucket while connected to your Bitbucket account.
And indeed when you switch to log in with an account or another you get a warning message that some feature will be disable, but you will always be able to switch between them an access both your organizations. You could also grant access to your Github or Bitbucket account to both of your SonarCloud organizations by adding them as members of your organizations.
I hope that helped.