Hi
No, Bluetooth does not have to be on at all times on your iPhone for the watch to be connected to the paired iPhone or to use certain Internet-based services.
When your watch is unable to connect to the paired iPhone via Bluetooth, it will attempt, as a fallback, to connect to a known, compatible Wi-Fi network.
When connected to Wi-Fi, the watch still offers a number of features. If it is unable to connect either via Bluetooth or Wi-Fi, it again still offers some - although fewer - features:
How to use your Apple Watch without your iPhone nearby - Apple Support
For the best performance from your iPhone and Apple Watch, including the most power-efficient communication between them, Apple recommends keeping both Bluetooth and Wi-Fi permanently enabled on your iPhone.