As mentioned in the first response above, it can take some time for Google to find a new website and then more time for it include results from that site in its index.
The .com and .org.uk sites are treated as completely separate entities by Google, so there's a danger that your old site will continue to rank higher than the new one because it has been around longer.
To resolve the situation I would do the following:
First, install Google Analytics (GA) on your new site. You'll need this later and it's best to do it before you move on to the next step.
Second, add the old .com domain name to your .org.uk Squarespace site while ensuring the .org.uk remains as the default domain. This is achieved by changing the DNS settings for your old .com domain as per instructions in Squarespace admin system.
This means that your old site is no longer accessible and people trying to access .com will be redirected to .org.uk
This is only part of the solution because the page names will be different between the two sites and visitors could be directed from search results to a 404 - Page not available error.
Third, look at the landing pages to your new site in the Google Analytics site you set up. 404 errors will show up as page names from the old site. If any of them are getting significant traffice you can set up URL mappings . These URL mappings should point from the page url of the old site to the most relevant page url on the new site
For example:
/aboutus.html -> /about-us 301
/oldurl -> /newurl 301
When I've done this for some large site migrations we often set up a lot of URL mappings in advance and then check Google Analytics for any that slipped through the cracks.
Depending on the level of inbound traffic to the old site from search, links on other sites & blogs, social posts, etc you should be able to set up all the URL mappings you need by initially checking Google Analytics hourly (for busy sites) or daily for most sites. After a few days I would then switch to checking once a week for a couple of weeks. That should trap all the 404s that are getting any significant traffic.