After installing Lightroom Classic v 15.4.1 and updating catalogue, I can no longer open LrC.

This happened to me when I upgraded the catalog to V 15.4. The conversion went fine, and the catalog opened. However on the second attempt to start LrC, nothing came up on screen. In Win11 Task Manager, I could see that LrC was active in CPU and Disk activity. I left the computer for dinner and when I came back a few hours later, LrC was up and running. Since then, performance has been excellent. The catalog loads faster that previous versions.

Once you click on LrC to start it, just leave it alone. It might take hours to complete, but it is working on the catalog and will come up normally eventually.

This might be the same issue I and aa few others have described. Launch Lrc 15.4.1 and nothing happens. There is no splash Screen. But LR is running with very low CPU usage and low memory usage. It can be terminated but this will repeat. When LR is running but not visible, I noticed that the lrcat-wal file is often 2000K-4000K.
Solution for me is to load from a backup. But it could happen again.

Unfortunately you haven't given us deteiled informations about your issue and any informations about your system. Therefore, all proposed solutions are a little bit of guesswork.

Do you get any error messages? If yes, please post the messages here.

Does the update of the catalog work without error?

First try a reset of the preferences of LrC:  How to reset Lightroom Classic preferences (adobe.com)

It's recommended to backup your preferences before you reset the preferences to the default settings: https://helpx.adobe.com/lightroom-classic/kb/preference-file-and-other-file-locations.html

If this doesn’t help create a new catalog and see if LrC starts.

Hold the Ctrl-Key while starting LrC. This opens a catalog selection dialog.

Click on “Create a New Catalog” to create a new one and see if this catalog works. If yes, you could then try to import your current catalog into the new one.

If this also doesn’t help, please provide a copy of your 'System Info'. This can be get from the Lightroom menu item Help > System Info. There's a copy button in the System Info dialog. Press this button and paste the info into your next forum post.