Why do the Russian and German characters not speak in their respective accents?

It was a stylistic choice by the director. In this interview with Stephen Lemons, Jean-Jacques Annaud makes it abundantly clear that as far as he's concerned, once you're immersed into the story, the accents are largely meaningless and may even detract from your enjoyment of the film. He also highlights some movie classics that followed the same path:

Some journalists have been critical of the fact that you shot the film in English, and that the main players, with the exception of Ed Harris, are British and have British accents. What's your response to that criticism?

Half of the market is an English-speaking market! If you give them actors who cannot speak English, it just doesn't play. And as a Frenchman, I can only direct in French or English; I cannot direct in Russian. There's no way you can do this movie in the Russian or German. You have to go with the original version in English. After that, you've got the choice of British, American or maybe Australian actors. I remind people that movies are made in the language of their audience. When Shakespeare did "Romeo and Juliet," he didn't do it in Italian, or even using English speakers with Italian accents. This applies as well to "Dr. Zhivago," which was set in Russia, but had English actors. It takes about five or 10 minutes to accept it, but once you're in the story, you forget that those people are English or American.

and in this interview with Ed Harris in Entertainment Weekly:

”[Director] Jean-Jacques Annaud specifically asked me not to do a German accent. A lot of times the German accent in war movies becomes a cliché – a ‘Hogan’s Heroes’ kind of deal. But I worked with a dialect coach and did a Middle Atlantic thing, a little more cultured than I myself am. A little more refined, if you will. At least I didn’t sound like I was from New Jersey.”