App not available on the App Store
The developer has most likely removed the app from sale from UK/all the App Stores.

You can check with the developer about the availability status by finding their contact information on the app's website. If you do not know the website address, try doing a Google search and see if you can locate it.

If the app is still available on App Store other than UK, you can install it by creating a different Apple ID and set the country as the one where the app is available. (This should be pretty easy to do if the app is free, as you won't have to setup payment).

Update: As mentioned in your response, if you are unable to locate the website via Google search, the most likely explanation is that the app is now defunct.