Translation of a element['#object'] rendered field I have some fields in which I rendered additional fields from a nested term via element['#object']:

e.g. {{ element['#object'].field_country.entity.name.value }}

Unfortunately, this fields do not translate inside the node view although the node and the nested term itself is translated.

How can I make them display with their translation?
Use getTranslation(), see How to load translated entity values in Twig for SEO purpose?.

In your case you can try something like this:

{% if element['#object'].field_country.entity.hastranslation(element['#object'].langcode.value) %}
  {{ element['#object'].field_country.entity.translation(element['#object'].langcode.value).name.value }}
{% else %}
  {{ element['#object'].field_country.entity.name.value }}
{% endif %}