I say conquest by the Romans/Christianisation, because the inhabitants of the province of Britannia presumably evolved to worshipping in a Roman way first, while the Irish went straight to Christianisation.
And to what extent was it similar across the whole British Isles?
Firstly, FYI - the term 'British Isles' is highly controversial in Ireland. The Irish do not consider it to be a simple, innocuous geographical term. It was introduced into the English language in Tudor times by a Tudor courtier as a political expedience when the English Crown governance of Ireland was under challenge.
The Irish government issued a statement in 2005 because of the growing concern that the government had over its modern day usage:
"The British Isles is not an officially recognised term in any legal or inter-governmental sense. It is without any official status. The Government, including the Department of Foreign Affairs, does not use this term.
Our officials in the Embassy of Ireland, London, continue to monitor the media in Britain for any abuse of the official terms as set out in the Constitution of Ireland and in legislation. These include the name of the State, the President, Taoiseach and others."
Anyway to get to the rest of your question. There is a body of work available to Irish scholars on the pre-Christian Irish world. Christianity arrived in the Ireland around the 5th century AD. The ancient Irish texts written in the Irish language between the sixth and twelfth century tell much about that world and the pagan religion - the Leabhar Gabhála, or Book of Invasions, describes the coming of the Gaels to Ireland and the pantheon of Irish Gods and Goddesses known as the Tuatha dé Dannan or the tribe of Danu, the Goddess. From many textual sources the religion is described as are the main feast days: Samain on Nov 1st, Imbolg on Feb 1st, Beltine is May 1st and Lύnasa on Aug 1st. The most important of these was Samain. It was the New Year in ancient Ireland and Oíce Samain [the Eve of Samain] was later appropriated by Christians and changed to Hallows' Eve or Halloween.
The main religious ceremonial sites are also known, Tara [in modern day County Meath] being the central one. The ancient Irish law system, Brehon law, also describes life in pre-Christian Ireland and these texts also describe that ancient word. We know that there was an interdisciplinary connection between the druids [the priests of the ancient religion] and the Brehons, the lawmakers. Brehon law remained customary in Ireland until the early 17th century.