Yes, obviously it's legal in England, or the coins wouldn't have been found, reported, and ultimately sold, as per the article. It's also legal in Ireland.
There are prohibitions and licensing requirements in both countries for search of heritage sites, national monuments, and other protected sites, and reporting requirements for unintentional "heritage" finds.
There are prohibitions and licensing requirements in both countries for search of heritage sites, national monuments, and other protected sites, and reporting requirements for unintentional "heritage" finds.