A few quick background bits of history before we begin. The 1812 Overture was written in 1880 to commemorate the Russian victory over Napoleon during his invasion of Russia. It was originally written as part of the festivities for the opening of the Cathedral of Christ the Saviour, also built to commemorate that victory. The Tsar would also have his 25th anniversary of coronation in the same year, and an All-Russia Expo was coming to Moscow, so that idea was that the piece could be used as a big part of the spectacle for these events.
I say that because the piece has become overwhelmingly associated, at least in the US, with a completely different event: 4th of July. And this association is actually one of the biggest reasons why the cannon fire is so well-known today.
I'll get to that in a minute. First, did Tchaikovsky ever hear the piece with the cannon fire as he intended?
The original idea not only included very precise cannon fire, but an additional brass band along with the pealing bells of the cathedral and other churches in the area. The cannons were going to be set up to electrical switches that would set the off at precisely the right time. This was a very complicated undertaking, but hey, the Tsar wants a big celebration, the Tsar gets a big celebration.
Except, Tsar Alexander II was assassinated in 1881, before the planned celebrations. This, along with the fact that the cathedral didn't finish on time, meant that the piece was premiered in a tent at the Expo, with no cannons or bells.
It still made an impression, though, and became one of Tchaikovsky's most well-known pieces. When he went on conducting tours it was usually on the program, most famously at the dedication of Carnegie Hall in New York, which he led in 1891. However, it seems that none of these performances, nor most of the performances in the decades after it was first premiered, featured the cannon fire as written in the score.
Part of this was practical. I am not a military expert, so someone who is could possibly chime in about 19th century artillery. There are 16 shots in a roughly two minute period in the piece. There doesn't seem to be any contemporary cannon that could be safely reloaded and fired this quickly, necessitating multiple cannons (and multiple crews). The electric switches in the original plan were devised to get around this, but that was a much more complicated proposition at the time than it would be today.
The earliest recording, from 1915, uses percussion instead of cannon fire, and this was the common practice for many years. Other orchestras used rifles firing blanks in their performances rather than the more cumbersome cannons. But the big breakthrough came when recording technology was advanced enough to allow overdubbing.
In 1954, Antal Dorati recorded the 1812 Overture with the Minneapolis Symphony. They then recorded a set of muzzle-loaded cannons at West Point, and overlaid those into the final recording. Herbert von Karajan did the same in 1966 with the Berlin Philharmonic.
So audiences were starting to get familiar with the concept of actual cannon fire in the piece. But to really cement it in peoples' minds, another great conductor and showman would make it part of an American tradition...
(in case it's not clear because of the edit, this continues in my reply to this comment below)
EDIT: As u/TheMusicArchivist points out, the cannons and bells in the original score are not necessarily a musical choice by Tchaikovsky. Since the original idea was for this to be performed as a part of a big celebration, the main point was more to make a lot of noise than for great musicality. In fact, Tchaikovsky was not a huge fan of the piece himself, saying that it was "very loud and noisy, but lacks artistic merit, because I wrote it without warmth and without love." It's worth repeating that he conducted many concert performances of the piece indoors without the cannon fire, which suggests that he didn't think it was a musical necessity.