I am using BGS WMS borehole data in QGIS:
https://map.bgs.ac.uk/arcgis/services/GeoIndex_Onshore/boreholes/MapServer/WmsServer
I'm trying to get a tooltip to come up displaying the URL from the attributes for each point.
How can I do it or are there any other options for getting the point to open a URL in the default web browser when clicked?
You can't, at least that's not how WMS is meant to work. You likely want WFS. There are a few questions about the differences: What are the differences between WMS, WFS, WCS, WPS?
Short version: WMS is a pre-rendered map - you never have the raw data locally. While you can get information about the underlying features via a GetFeatureInfo request, that's usually done by clicking on a feature, not simply mousing-over.
WFS is for the raw vector data. In that case you have the points locally and you can do what you like with them including setting up tooltips (whether that is possible, and how to do it I leave for someone else).
Unfortunately for you, the BGS Borehole data seems to be WMS only: https://www.geoseer.net/s.php?q=bgs+borehole
