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As long as your hamster eats a good mix then teeth shouldn't be a problem, many hamsters enjoy chews but they aren't essential, I can understand you wanting to keep things natural but a lot don't enjoy chewing on wood that much, another option is dandelion root but not too much or too often due to it's medicinal properties, whimzees do tend to be the favourite chew with most hams.
Most natural rough surfaces will help with keeping nails trim to some extent, you can use a granite tile in an area where they walk over it frequently or some stones, not smooth ones, you do have to be careful with arranging stones in the cage to be sure they are completely safe though.
As Razor mentioned the sand bath will help if your ham enjoys digging in it.