How accessible was classical music to ordinary people in the Georgian era? Was it just reserved for the wealthy and upper class?

Regency eh? Can I take a stab in the dark that you're writing some sort of bodice/cravat-ripper? Can you be more specific about your major plot point? And your character's class? "Ordinary" people is a tricky in a period in which the middle class as we know it didn't really exist. Regency is a bit late for my really thorough knowledge of the English music scene, but I can give you a stab at whether or not your plot is reasonable. I will cheerfully admit am a regular reader of books set in the Regency period with half-nude people on the cover so I'm probably the fairest opinion both as a historian and reader you're likely to get on it...

Access would be a limited, but this is a period when things are opening up, so you're much better than if you'd set the story in 1730, there are certain situations in which the not-poor and not-titled but let's say "comfortable classes" of late 18th c.-early 19th c. England would have a chance to hear art music. The first thing you can take a look at, depending on the class of your character, would be them attending a public concert. This is the time of the rise of the public concert, which is pretty close to what we think of as "going to the symphony" today, in terms of the social event, the price, the music presented, and the classes who usually attend. A ticket then would be more of an impact to a wallet then than today though. A subscription to a full season of concerts (9 evenings) in Bath in 1795 cost about 1 guinea, and a single concert about 12 shillings. Using back of the napkin math and a sketchy understanding of pre-decimalized British currency, that single subscription would cost about 700 pounds in today's British money. It's a little hard to compare because they didn't do tiered seat pricing for concerts back then, but a mid-range 9-event subscription to the London Symphony Orchestra (off their website I used 30 pounds a ticket x 9 tickets and the 20% subscriber discount) would run you about 220 pounds today. So it was pricier then, but not totally unaffordable. If your character say is a woman being courted by a man of decent means who liked music (or wanted her to think he liked music!) he could very well take her to a public concert.

One other option, and a perennial favorite setting of novelists in your situation, would be Vauxhall. The music at Vauxhall, however, tended to be very nativist/patriotic, especially during the war. About the only foreign music regularly on offer there was Handel, and he's like honorary English, especially without his umlaut, and he had a darn statue in the gardens to boot. It depends on what you're going for with exposing your characters to music. Vauxhall, however, was cheap as chips, and plausibly accessible to any character who can physically get there, dress decently, and scrape together the shilling for admission, which in 1800 is a big chunk of London. Its also pretty decently covered in historical record (like newspapers), you can reasonably recreate evening playlists for Vauxhall given a particular year. There was also a rival, Ranelagh Gardens, that you should take a look at, though it was more exclusive (meaning expensive.)

If you give me more details I can give you more books and ideas, but right now your question is too broad for an answer other than "read this whole pile of books on the consumption of foreign art music as a tool of class performance in late 18th century England." Which probably isn't what you're looking to do today, and pathetically I've already done it, so let me know. :)