Since Apple manages the whole thing through their own app and not a Goldman Sachs app, it should be fairly seamless as everything should look the same and you won't need to make a new login or worry about how to start paying a new bank. If there's new cards to be issued it's likely it will just show up in the Wallet app and they'll mail you a new physical one. Last year Apple moved the Apple Cash card from Discover to Visa and most people didn't even know that happened. There was even a button to switch it over sooner if you wanted to.
With the Savings Account I expect it will be similar as long as they can find a bank willing to offer a similar APY. Especially for people who just use it with the Apple Card and don't deposit directly to it using the routing/account number, you probably won't really notice.
source: all speculation, but I have worked extensively in payments for years and have launched banking products.