OP you're better off using an offline wallet of which you have exclusive control of your keys.
A fork doesn't necessarily mean two chains, but in case it does, you'll have coins on both of them.
I am not completely sure how the Omniwallet works, but I know you can store some other coins besides BTC on it, like MaidSafe and Tether etc. But what would happen if storing these coins on a bticoin Omniwallet address and Bitcoin would fork? Would that somehow "mess up" things and could you lose these other coins? I don't really know the techinals on how all this works so it has made me a bit confused and worried.
It's ok as long as you hold your private keys, but honestly, if you want maximum 100% certainty, just get bitcoin core and run a full node. And if you want to access the BIP148 chain coins, you will need to download an BIP148 full node, so you are going to be running 2 nodes for each chain in order to access both chains.
Do not expect to leave your coins in some exchange praying they list both coins, just get full nodes. It's annoying to run 2 nodes but that's life.
BIP148's goal is to force-activate SegWit, which won't happen because we'll get SegWit2x before then.