I found a couple of ways to achieve it, posting here, but I'm open to better/cleaner ways. I'm especially unhappy in my answers with the way to access the tag name - seems so overly complicated trying to get the raw name?!
Both code blocks live in a mytheme_preprocess_node():
Answer 1: create a new variable with all the tags in it:
<?php
  $allTags = [];
  $addTags = function($fieldName) use (&$allTags, $variables) {
    if (isset($variables['content'][$fieldName])) {
      foreach (\Drupal\Core\Render\Element::children($variables['content'][$fieldName]) as $i) {
        $allTags[]= $variables['content'][$fieldName][$i];
      }
    }
  };
  $addTags('field_colours');
  $addTags('field_tags');
  usort($allTags, function ($a, $b) {
    $tagA = $a['#taxonomy_term']->name->getValue()[0]['value'];
    $tagB = $b['#taxonomy_term']->name->getValue()[0]['value'];
    return strcasecmp($tagA, $tagB);
  });
  $variables['content']['allTags'] = $allTags;
Then in Twig for the node template: {{ allTags }}.
Answer 2: mix the tags from the colour field into the tags field in the render array
<?php
  $allTagsTemp = [];
  foreach (\Drupal\Core\Render\Element::children($variables['content']['field_tags']) as $i) {
    $tag = $variables['content']['field_tags'][$i];
    unset($tag['#weight']);
    unset($variables['content']['field_tags'][$i]);
    $allTagsTemp[] = $tag;
  }
  foreach (\Drupal\Core\Render\Element::children($variables['content']['field_colours']) as $i) {
    $tag = $variables['content']['field_colours'][$i];
    unset($tag['#weight']);
    unset($variables['content']['field_colours'][$i]);
    $allTagsTemp[] = $tag;
  }
  usort($allTagsTemp, function ($a, $b) {
    $tagA = $a['#taxonomy_term']->name->getValue()[0]['value'];
    $tagB = $b['#taxonomy_term']->name->getValue()[0]['value'];
    return strcasecmp($tagA, $tagB);
  });
  $variables['content']['field_tags'] += $allTagsTemp;
Then in Twig {{ field_tags }}.