Q: Does the 'Under Review' status for Springer mean that it has gone for peer review?

The exact meaning of a certain status update differs across journals, and Springer (Nature) has many journals. :-)

Now, you haven’t mentioned whether there were any status updates before ‘Under Review.’ If not, and it directly went to ‘Under Review’ after submission, this indicates the internal check (for aspects such as scope match, novelty factor, quality of the writing, and adherence to guidelines). If the manuscript clears the internal check, it will be sent for peer review. If however there were status updates before this one (such as for the internal check), the manuscript has been sent for peer review.

Hope this helps. For further clarity on status updates, you may refer to the following resources:

• Does "Under Review" mean the paper has been assigned to an external reviewer?
• Does "Under Review" mean that the paper has passed the editorial check?
• A guide to understanding journal statuses [Handbook]

All the best for the next stages!