It's hard to draw direct parallels. Music, it's meaning, and how we listen to it has been revolutionized since the late 18th/early 19th century. Still, what we think of as classical music would only have been one part of the music people would have been exposed to. It’s difficult to be specific because what else they would have listened to would have depended on dozens of factors, like where they lived, when they lived, how wealthy they were, etc. Still, there are better candidates for a pop music equivalent than the compositions of major composers at the time. I’ll stick mostly to the music of England and Ireland, as that’s what I’m most familiar with. Also, when I say “classical” here, I mean it in the broad sense, or what we generally think of as classical music, not the more rigid definition that would include Mozart but exclude Beethoven.

As far as “other” music people would have listened to, you have plenty of categories to choose from. Religious music is certainly a prime candidate. In the Anglican Church, hymn books were standardized by the 18th century so you’d have people across the kingdom listening to and singing from the same book of hymns. But I’m guessing religious music isn’t quite what you had in mind for “pop” music.

There was plenty of secular popular music as well. Story-telling songs, or ballads, had been popular for centuries. Francis James Child published the first real scholarly collection of ballads in 1860, although some songs can be traced back centuries before that. Many of the collected ballads can also be found across the British Isles with variations on the lyrics, or with different melodies, which shows these songs were spread far and wide in some form or another, often tweaked slightly to make more sense in each area they were performed.

Shanties, or work songs, were also prevalent, although it’s hard to call them pop music as they would mostly be heard while, well, working. Sea shanties are a notable example, and would have been structured around rhythms that’d help sailors synchronize their movements on a specific task.

Dancing is another rich source of popular music, both during the time of Mozart and Beethoven, as well as for centuries before and after, up until the present day. The British Isles are home to many different dancing traditions, from refined country dances to more impromptu social dances, to carefully choreographed regional styles popular among the lower classes. All of which frequently have a specific tune associated with each specific dance. If a dance was especially popular, you can bet that people would have heard the associated tune a lot too. There are many, many publications of dances, music, and tutor books going back centuries – all of which we can use to get an idea of what tunes people were interested in playing and learning. What’s more, we can look for specific tunes and dances in sources from across the decades (or centuries) to see which pop music remained popular.

As far as overlap goes, there was also plenty of that. Classical styles certainly influenced local music traditions. Take Turlough O'Carolan, Irish harper and composer of the early 18th century, known as the national composer of Ireland. Many of his compositions are in a distinctly baroque style, while others still are in the more traditional Irish harping style, representing an intrusion of classical music into the music tradition of Ireland. Staying in the world of Irish traditional music, we can see even more direct crossover by looking at what kind of music was published in collections of Irish traditional music. Looking at O’Farrell’s pocket companion for the Irish or Union pipes, being a grand selection of favorite tunes both Scotch and Irish…, published ca. 1836, we find not only a collection of Irish and Scottish traditional tunes, but also a number of classical Baroque tunes arranged for the Irish uilleann bagpipe. Among these are a number of Handel compositions, including “The Arrival of Queen Sheba.” As you may have noticed, this book was published a century after Handel and his music was published and most prolific. This suggests that certain bits of music from the classical world directly jumped over to the pop or traditional world. Folk crossed over into classical music as well, with Hyaden being a notable contemporary example of a baroque composer drawing on folk tunes for his compositions. As you can probably tell, I’m not very well versed in the classical side of things, so I can’t elaborate much further than that.