I finally got the name of the SMS platform/service provider from a helpful account manager at Mblox.
The company, Collstream, quickly replied to my email given me details of the company that appears to have sent the unsolicited SMS. The name of that company is CPN Ltd, who run a website called ‘claimsjustice’ (dot co, do uk) … I’m not putting the URL in here to prevent any benefit from linking to their site.
I have sent them an email pointing out the various ways that I believe they have breached the regulations. As soon as I get a response I will post the details here.