Callouts to web services started failing with CalloutException: Server chose TLSv1

This corresponds to Salesforce disabling TLS 1.0. Pointing https://www.ssllabs.com/ssltest/ at the publicly accessible web services returns the following result:

As a short term fix while upgrading the web services to support newer versions of TLS, you might be able to deactivate the corresponding critical update:

Update Name

Require TLS 1.1 or higher for HTTPS connections

Update Summary

This update strengthens HTTPS connections to and from this organization by disabling the older TLS 1.0 protocol and requiring TLS 1.1 or higher in HTTPS connections.