Initially, there was no indication this was a scam, they quoted back the emails to EON Next mentioning the final readings, in addition to the name of the landlord I was handing the account over to.
They wanted confirmation of my new address and date of birth, which I gave. Luckily, I didn't go any further before checking the number and doing a Google search:
0115 762 9950
There wasn't much information but one website signalled it as a scam so I hung up immediately.
I phoned EON Next to confirm whether this was/wasn't them. They confirmed it wasn't and recommended blocking the number. I asked for them to escalate this issue as there appears to have been a very recent data breach where emails to and from EON Next have been hacked. I wanted to warn people to make sure others are not tricked in the same way.