Based on what Luke wrote in  and my discussions here are my thoughts about the situation with anti-BIP148 exchanges. If you've no clue what UASF BIP148 is, read .
Disclaimer: this post probably likely contains unintentional mistakes and/or errors. Feel free to point them out. I recommend that you don't read only the post – make sure to also skim through the comments to find comments that highlight obvious and less obvious errors.
MYTH: MY EXCHANGE SAYS THEY PLAN TO STAY NEUTRAL
No, they won't stay neutral unless they add the option to allow you also access BIP148 BTC in your wallet when the chains split.
Exchanges can either support both "legacy BTC" and "BIP148 BTC" (that would be neutral, but when they say "neutral" that's not what they mean) or be hostile. When then tell you they're neutral, they're in fact hostile.
As Luke-Jr explained in , to stay "neutral" (hostile) is to remove a choice. If you don't have a choice, the only way to completely avoid risk is to avoid using your exchange while the chain is split. That doesn't mean just trading but extends to making deposits or withdrawals while the chains are split.
It is important to realize that it does not matter whether you "support" UASF BIP148 or not. If you don't have a choice, you're exposed to risk because your exchange doesn't want to protect your wallet.
PLAN A: MY EXCHANGE WON'T SUPPORT BIP148 BTC, SO I'LL SIT IT OUT
You can certainly do that, but in case you change your mind after August 1 and the chains are still split, you will have access only to "legacy" BTC. And sending them may be risky (see .)
Some exchanges have stated that they don't "recommend" to use their services while the chains are split. What does that mean? What if I trade anyway exchange and my trades later get invalidated? What if I want to deposit some BTC and my transaction gets reorg'ed? It seems that BIP148-hostile exchanges would have to completely suspend BTC-related operations from August 1st until the chains merge. In any case their customers should ask whether they'll be compensated in case they use their BTC accounts against the exchange's recommendation. I doubt that.
Or you can execute PLAN B: move your BTC to a customer-friendly exchange or wallet (such as Electrum, Mycelium, Samourai) that lets you retain full access to your BTC and even on both chains (if you "split" the coins). You'd have to do this before August 1, of course.
During the split, the situation may change from day to day or even hour to hour. The price of both "legacy" and BIP148 BTC may drop or rise dramatically.
MYTH: IT WILL BE OVER IN NO TIME
Maybe that is true, but there's no way to be sure of that.
Something that occurred to me yesterday is that if the BIP148 chain survives Day 1, it may very well survive much longer – several days or several weeks. As long as BIP148 BTC has a non-negative value, money can be made mining them.
In PLAN A above I mentioned that some exchanges may need to effectively shut down their BTC-related services, not just "recommend" that you don't use them while the chains are split. If that happens and the BIP148 chain survives, exchange services may need to stay frozen (or available under "use at your own risk" clause) for weeks. All businesses who accept Bitcoin payments but resist BIP148 would also be at risk during that time.
WHAT TO DO
Both pro-BIP148 and anti-BIP148 users should not accept this fake neutrality. Wherever you stand on BIP148, there's no reason to give up full access to your bitcoin wallet while the chains are split.
Get your exchange to add BIP148 support or have them provide the details (including whether they'd compensate you for losses due to their hostility) of their plans for what may turn out to be a multi-day or multi-week chain split. They have more than enough time to do that – for example the group behind the NY Scaling Agreement can develop, test and ship their code in less than 8 weeks .
If you can't obtain meaningful guarantees, consider PLAN B: withdraw BTC to a secure offline wallet or move them to a neutral exchange that intends to support the both chains.
If you want to learn more, read UASF Guide and check recent UASF BIP148 articles from this subreddit.
 https://UASFGuide.com – User Guide to UASF BIP148
 https://np.reddit.com/r/Bitcoin/search?q=bip148+uasf&sort=relevance&restrict_sr=on&t=month – UASF BIP148-related articles on /r/bitcoin
 https://imgur.com/a/a2oPs – FrankenSegWit – your next Bitcoin software: from Alpha to GA in less than 60 days!
 https://www.reddit.com/r/Bitcoin/comments/6dos0r/uasf_is_an_economic_boycott_against_miners_who/ – WTF is UASF BIP148?