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What’s the recommended way to update the theme without losing theme customizations?

If you made the customizations only in the theme options then you can update the theme without any problem, because all settings are saved in the database. Only if you made changes in the theme files/PHP files you have to take care. The recommended way is to download all themes files via FTP and save it. Then unzip and upload the updated theme into the theme folder (wp-content/themes/), always overwriting the old theme files. No worry, all settings, uploads, pages, and posts are saved in the database.

Usually I use the automatic theme update functions built into most other WordPress themes but Londra doesn’t show a button to update itself.

This function is only available for themes from wordpress.org