There are so many possibilities in English, and most of them have already been mentioned. I agree with the consensus that the concept is not nearly as familiar as a Christmas hymn or carol, either in German or in English.
I would just add one more I don't recall seeing mentioned in this discussion, namely, 'Easter carol.' Although it's probably the least common of all the options, there might be a very occasional context where it would fit.
Easter hymn - A traditional classical piece, with verses, sung in church from a hymnal by the congregation. Often called a chorale in German, especially if in the Lutheran tradition or the style culminating in J.S. Bach.
Easter anthem - A traditional, slightly longer classical piece sung in church by the choir, usually with organ.
Easter song - A simple, secular-style piece, like a pop song or children's song. In church, perhaps (1) a modern, easy praise song, e.g. with guitar and electronic amplification; or (2) in wider society, something like a secular song for children about the Easter bunny (if one existed, which I'm not actually aware of).
Easter chorale - Normally in English, a four-part hymn set by J.S. Bach or another composer of that era in the German (Lutheran) style, esp. in the context of a larger work such as a passion or an oratorio. (But since, most notably, the Bach passions end before Easter, most of the chorales known to English-speaking audiences are not Easter chorales.)
spring / Easter carol - A spring or Easter song in the style of a traditional English or European folk song, not necessarily religious, with verses and often a refrain. See the [old, red] Oxford Book of Carols or the [complete, unabridged, not 'Shorter'] New Oxford Book of Carols, ed. Keyte and Parrott. These are much less common and well-known than Christmas carols, but some of them do exist.
There are also several specific liturgical musical terms that can be combined with 'Easter' -- an Easter chant, sequence, responsory, motet, etc. Some of them are the basis for later pieces such as hymns or anthems.