Exception:

We can’t upload your photo right now. Please wait a few minutes and try again.

Resolution:

1. Go to Salesforce Setup.
2. Go to Sites in the Setup.
3. Select the Site.
4. Click the Edit button and enable the check box “Allow Access to Standard Salesforce Pages”.
5. Click the Save button.

Workaround if “Allow Access to Standard Salesforce Pages” cannot be enabled:

Use custom component using setPhoto() method.

https://developer.salesforce.com/docs/atlas.en-us.apexref.meta/apexref/apex_ConnectAPI_ChatterUsers_static_methods.htm#apex_ConnectAPI_ChatterUsers_setPhoto_2