Due to a known iOS/WatchOS issue that Apple is working to resolve, your iPhone needs to be open when you wish to initiate a new sleep session on the Apple Watch. We understand that this is somewhat troublesome and we are monitoring the situation. As soon as progress has been achieved, the restriction will be lifted.
In the meantime, to start a sleep session on the Apple Watch, please:
Open Pillow on your iPhone, keep it open and do not press any buttons
Open Pillow on your Apple Watch
When you are ready to start a sleep session, select (one of) the preset mode(s) you have created and tap on "Start"
Please make sure to lock the screen only after a sleep session has started on your iPhone to avoid unexpected shutdown of the app during the night.
Just to clarify, Pillow on the Apple Watch should require the iPhone only if you try to start a sleep session with either of the following options checked:
iPhone speaker
iPhone vibration
Sleep aid
Audio recordings
These options use the iPhone, so connectivity is required in order to start a sleep session. Once the sleep session starts, you can lock your iPhone and place it on the bed-stand.