Yes and no!
I'll try to explain:
The Irish C&C group (which includes Bulmers Ltd of Clonmel) have the right to use the Bulmer's brand name within Ireland only. They use the name to manufacture Bulmer's Irish Cider. When they decided to market this product internationally, they had to find another name and 'Magners' was born. (i.e. Magners is the same drink as Bulmer's Irish Cider). So that's the 'yes' part of the answer disposed of.
However, a completely different company, H P Bulmer Ltd of Hereford, have the right to use the Bulmer's brand name in the UK (e.g. Bulmer's Original Cider). Possibly to avoid confusion, they now brand their products as Woodpecker, Strongbow & Scrumpy Jack. So the disappearance of the Bulmers brand name in UK pubs has no direct connection to the emergence of the Magners brand. (They're completely different drinks, produced by different companies). So there's the 'no' part of your answer.
Chris