The local (Northwest) pronunciation of these is clem-AT-is and Pe-ON-ee.
However beig a long time gardener, I pronounce them the other way: CLEM-a-tis and PEE-on-ee.
When I get confused on pronounciations, I look it up and check several sources. I always go with the horticulturists.
Here they say wy-GEL-a and I say (or am trying hard to say) wy-JEE-la, as well.