Not directly from privatisation, but in 2006 BT was forced to split the infrastructure part of their business into a separate company (OpenReach). They were forced to allow other providers to use their infrastructure, which during the broadband-era led to pretty healthy competition.

I'd say the issues today (fragmented services offerings, high prices, low speeds) are more due to under investment in the infrastructure (keeping copper for too long) - which could have just as easily happened if it was government run.