Out of curiosity, I signed up for this service. It looks like there's no option to export your private keys from this wallet, and it looks like an online wallet (in which they control the private keys for you).
Your best option is to move them out to a wallet from which you control your private keys and can export them if necessary. If you read
theymos' thread about this, you will find some key wallets which allow this.
If you're unwilling to move your coins out of your Luno wallet, you can always contact their support and find out whether they are going to support Bitcoin Cash and any other splits on August 1st. It's likely that they will not be able to provide a good UI for you to use these coins, so they won't bother giving you access to those chains.
Even if they tell you that you
will receive your coins, you will need to trust that they will follow through with this.