For instance, Bach's Toccata and Fugue in D Minor, would this have been played by organists throughout Europe at the time? Or did the spread of composers' music accelerate at a later date?
Mozart traveled A LOT, even for the standards of our time. He also published his music, he was known in many countries.
Beethoven's works were also published in different countries. His 9th symphony was commissioned by the Philharmonic Society of London in 1817. He was known even across the Atlantic.
Most of Bach's music was not published during his life. He was nowhere near as famous as Mozart or Beethoven.
Musicians did travel during the Baroque period, and there were some super stars touring all over the big cities. Publishing was a good way for music to spread, and we find music traveling across Europe even before the Baroque, but it was nowhere close what was seen during the 19th century.
Musicians did come in contact with music from other countries, and even traveled to pick up some things and use them in their own music.