You can also do this in the Field Calculator with the refFunctions plugin. You can use the Field Calculator to update a layer geometry as well as fields. refFunctions gives you a "geomdistance" function to find the nearest line within a given distance(or "geomnearest" if you don't want a threshold) and will return an attribute or the geometry, and the "closest_point" function will find the closest point on a given geometry. String them together like so to calculate new geometries for your point layer:
closest_point(geom_from_wkt(geomdistance('snap_lines','$geometry',10)) , $geometry)
Instead of directly updating the geometry you can calculate a field with the snapped geometry instead. I store multiple geometries for snapping culvert points to different stream layers and I can easily update the point geometry in the Field Calculator depending on whose stream lines I need to use.
There are some limitations with this, both layers need to be the the same CRS and the geomdistance function will give you an error if you have more than 100,000 points but you can change this limit if you edit refFunctions plugin file.