The price cap for the changeover date quoted was that from 1 October 2020 to 1 March 2021.

The price cap data is in this link. You'll notice that it doesn't set prices as such, just the maximum amount chargeable at nil kWh (so effectively the maximum standing charge) and 4,200kWh. Hence both the nPower and Eon Next tariff prices could comply with the cap. The OFGEM data is VAT exclusive.

Default tariff cap level - 1 October 2020 - 31 March 2021 (ofgem.gov.uk)

Multi-Register Metering Arrangements are more complex than single tier charging and allow some flexibility on the rates charged by suppliers ( so different suppliers can meet the price cap in slightly different ways.). However, if the promise was for your prices to remain the same upon the transfer to Eon Next and in fact they didn't AND this resulted in you paying more, then you have grounds for complaint. If you have your consumption data between 11th December2020 and March 31st 2021 the difference in the resulting bill could be easily calculated. I would suggest that you obtain this information from Eon Next if you don't already have it.

From 1 April 2021 the new price cap would have been introduced and then only Eon Next's prices would be relevant.

But right at the beginning of this thread you stated "When E.ON Next took over nPower they sent an email saying that there would be no immediate change in tariff; there might be a change when the price cap changed in October."

On their FAQ page they said this

"Will my tariff remain the same?

If you’re on a fixed tariff there’s no change to your prices or tariff end date. Variable prices may change after the regular price cap review, as they do now."

Some would argue that those words leave a degree of wiggle room.
Current Eon Next and EDF customer, ex Zog and Symbio. Don't think dual fuel saves money and don't like smart meters. Chronologically Gifted. If I offend let me know by private message, but I’ll continue to express my opinions nonetheless.