Go to www.openstreetmap.org.

Search for the river and, hopefully, there will be a relation grouping all the ways that conform the river.

With that id, do:
writeCoordinatesFromRelationId.sh name relationId

and a file with the node coordinates named name.id.coords will be written.

Alternatively, if you have nothing but a list of ways, you can use
writeCoordinatesFromWaysIds.sh name wayId1 wayId2 ...

The output will be a file with node coordinates named name.wid1.wid2.coords

If those ways are too many, check that you get them with:
getWaysIdsFromName.sh search_words

If so, just do:
writeCoordinatesFromWaysIds.sh name `./getWaysIdsFromName.sh search_words`

Packages needed: wget, xgrep

