Reeds (typha, or reedmace) and rushes (bulrush) are often confused, although they belong to different botanical families. In particular, typha (reedmace) is often misnamed as bulrush. Since both plants grow in water, you will often see them intermixed in streams and along riverbanks. So, a "rushy bed" is simply a bed of bulrushes (or typha).
Note also that a reed is a rustic musical instrument. So the reed in the poem would be made from the stem of a bulrush (or more properly, reedmace or typha), taken from its rushy bed.
Typha, or reedmace (Wikipedia)
1.4 (literary) a rustic musical pipe made from a reed or from straw.
‘as if thy waves had only heard the shepherd's reed’
Oxford Online Dictionary