Many thanks for all the suggestions!
I tried everything but nothing worked. Looks like I misunderstood the upgrade and product key validation process and got rid of my Win7 Starter before actually upgrading the key, so there was no way to recover from that.
But it also could have been a malfunctioning (outdated) hardware element (according to the MS support person I talked to) that just prevented MS from validating the product key.
In the end I went with Ath's suggestion and tried to find out how to validate it manually via MS support.
I followed the process described here:Activate Your Windows 10 License via Microsoft Chat Support
First it put me through an automated service that couldn't validate the product key. Then I talked to a customer support person who also couldn't validate it through her system.
So I was put through to technical support in India, over a terrible quality phone line, where I spent an hour and a half trouble-shooting with an otherwise very helpful technical guy.
He had to remote into my machine eventually to carry out the upgrade. As far as I can tell they just gave me a new product key for a slightly different version of Win 10 Home. It didn't look like they managed to truly validate my old one.
In the end it wasn't for free, as I spent 2 hours on the phone getting this sorted (and time is money, right?). Still, I'm impressed with the huge resources MS seems to be throwing at the support end.
I didn't have to wait at all to get through, and then the guy was willing to stay with me for almost 2 hours over a toll-free number, and even called me back to check whether it's all worked out, while he was telling me that they are getting a huge volume of support calls of this nature.
It's all hunky-dory now, and I'm glad I managed to rescue this old kit, so at least I got one Win10 machine in the house to experiment with.
Thanks again, everyone.