I think fine-grain basalt is the next best thing. Folk don't seem to have put the same work into polishing them as they did with greenstone though. Don't know if that's because it doesn't polish as well or if it's regional differences.
Any pics I've seen have had a similar cutting angle to greenstone, maybe just a touch more obtuse and rounded.
If it helps any, here's a couple of pics of greenstone. It comes in various grain(well-particle) sizes.
Finer material is knappable but doesn't polish up as well. The bottom-right lump is an old broken drill of really tight stuff.
Larger grained greenstone is a bit brittle and prone to endshock. Not good for an axe but folk would have known the design changes to make. The large grain material was heavily mined, so presumably was heavily used).
Weathering is present even on artifacts and faces only 1000 years old or so. Weathers to white or purple depending on particle size(purple is large). Weathering is never green
.
The dark medium grain greenstone polishes up to a very useful edge but I've only tried it on that one wee 1 1/2" "pocketknife".