I have recently set up search console, however the dashboard preview of my site shows the old home page, rather than the new redesigned home page. It has been at least a few days. Will this eventually update or have I done something wrong where it is not seeing the new site? Sitemap has been updated.
Google shows the old home page from it's cache, last time when it crawled your site it still had the old design. You need to wait until it crawls your site again and caches your new design. It depends on many factors when will it crawl your site again, please check this article about it: https://support.google.com/webmasters/answer/34439?hl=en
You can check how google sees your web pages using the fetch as Google tool in search console:
Hope this helps.