Thank you for reaching out and bringing this to our attention.
If the favicon isn’t appearing in the Google search results, then it’s possible that Google still hasn’t reindexed your store since you’ve made the changes. Though Google will automatically recrawl your site periodically and the issue may resolve itself in a couple of days, you can also submit a sitemap or ask Google to recrawl your URLs if you want to speed up the process or if the favicon doesn’t appear after a few days.
That being said, I’ve taken a look at your store (covertathletic.co.uk) and it looks like you have the password protection for the online store enabled. While this shouldn’t cause any issues for the favicon, it may prevent Google from indexing some of the other pages from your store and/or prevent the Facebook team from accessing and reviewing your products (for Facebook and Instagram by Meta).
Have you considered using something like Shopify Inbox to interact with customers live on your store or provide an inbox for customers to leave messages? Do you have a marketing plan prepared for your store?