|Black and tan|
The most common one was probably bitter with brown ale. We called this 'brown and bitter' when I was student in Warrington, but later when I worked in Liverpool a colleague firmly told me that it was called 'brown bitter'. When I asked why he chose it, he pointed out that you got well over a half of bitter along with your brown ale: this is the same mentality as people who get pint glasses in beer festivals but order halves, thus getting overgenerous measures from lazy bar volunteers.
Another mixture I came across was light and bitter - a bottle of light or pale ale with the bitter. This was sometimes misheard by bar staff in noisy pubs as mild and bitter. I have wondered whether these mixtures arose as an attempt to disguise poorly kept draught beer in the 50s and 60s by diluting it with bottled beer, which would at least have the virtue of consistency. Other combinations I came across included (using the Liverpool terms) 'mixed' (bitter and mild) and 'brown mix' (brown and mild).
'Black and tan' was bitter and Guinness, although I've seen suggestions it should be a pale or light ale rather than bitter. This was often poured to create two layers, as in the picture. I've heard the term used to describe dark mild and Guinness, although using the word 'tan' as part of the description of a mixture of two dark drinks seems odd to me. Light mild would make sense, but that wasn't readily available where I used to drink. The term 'black and tan' was never used in Ireland for obvious reasons; they prefer to call it 'half and half'.
'Snakebite' was bitter and cider in the 70s, but more recently I've heard the term applied to lager and cider. This was quite a lethal mixture, or at least it seemed so at the time. Guinness and cider was called 'poor man's black velevet', the original of course being Guinness and champagne. Once for my birthday party I made a black velvet punch which was very popular, although I replaced the champagne with Pomagne. If I were to try it again today, I'd probably use cava.
There were other mixtures, such as lager and lime, lager and blackcurrant, Guinness and blackcurrant and cider and blackcurrant, but these were just ways of sweetening beer for people who basically didn't really like the taste of it. Also, obviously, they are not mixtures of two alcoholic drinks.
Nowadays I very rarely hear people ordering such mixtures. Perhaps draught beers are more consistent and interesting nowadays. Price too may be a factor, bottled beers being significantly dearer than draught. Or it could just be that the times they are a-changing.