The closest thing to “freedom of speech” that exists in the EU is article 10 of the European Convention on Human Rights which includes the “right to freedom of expression”. This is subtly distinct from freedom of speech (which is a very USian idea). The article goes on to say that the exercise of this right can be restricted for various reasons including “the protection of health or morals” and “the protection of the reputation or rights of others” (among others).

In terms of privacy, there’s no specific right to privacy in the ECHR, but the subtly different “right to respect for … private and family life, home and correspondence” including in Article 8. Again, this can be waived for the same reasons as above. These two articles often get balanced against each other, for obvious reasons. It also isn’t hard to see how this “right to be forgotten” could be covered under (or related to) this right to respect of private life etc., when it wouldn’t necessarily be covered by a simple right to privacy.

From what I have observed the US holds the idea of freedom of speech much higher than it is held elsewhere, particularly in Europe, where other “rights” (including privacy, protection from defamation and prevention of hate-speech) often take precedence. While in some case this can go too far (such as the Twitter Joke Trial), it is part of how European culture seems to operate. As such, while there is an obvious conflict with freedom of speech in this proposal, it may not be quite as noteworthy as it may seem from an American perspective.

With regard to the ECHR and the UK, (iirc) the UK was the first country to sign the convention (and the first country to be taken to Court by it) and it has been binding on the state since 1953. In 2000 this was strengthened through the Human Rights Act 1998 which made it easier for cases etc. to be considered with regard to the convention.