Never underestimate the requirement for a pint at a wake!

However, the reality is a bit different. A 'pub' is a Public House, and as such, has historically had a greater role in society than a mere bar and place to get a drink. Public houses could be used for a variety of state functions and, if necessary, still can perform that role. For example, Americans may have recently heard of Ralph Lauren's niece being brought to court... the court taking place in the local pub. (As a tangent, the Circuit Court in Ireland was so called because the judges would perform a circuit, holding court in various towns, so those who couldn't get to the fixed (district) courts could still access the law. These courts being held... yep... in the pub)

http://www.telegraph.co.uk/news/uknews/crime/10557028/Designer-Ralph-Laurens-niece-in-court-in-Ireland-over-alleged-air-rage-incident.html

Most specifically to your question, however, you want to look at the Coroner's Act of 1846. This mandated that bodies of the deceased be brought to the local public house for storage until further arrangements could be made. This was done because pubs tended to have cool cellars which would slow decomposition. A lot of pubs still have marble tables in the cellar for autopsies. As a result, as a matter of convenience, the local publican would also be the local undertaker.

The law mandating the use of pubs for storage of bodies was removed from the books in 1962 in Ireland (I cannot speak to the UK). This topic came up in conversation last time I was in Dublin, and the gist, which I have not confirmed, is that although it is no longer mandated to bring the deceased to the pub, there is no law prohibiting it, and it can still be done if, for some reason, transport or access to a morgue is not available. As you can imagine, also, the amount of publicans who are also undertakers has likely fallen.