If I have remote stations reached by a BYOC Cloud or BYOC Premises External SIP Trunk with UUI enabled on the trunk, will the other end of the trunk receive the UUI data?

That depends on what type of telephony device you have on the trunk. If you have a PBX or an SBC on the other side of the trunk, you can receive the UUI data. However, if your remote station is reached through the PSTN, it is unlikely the UUI data will persist through to the station.