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Nietzsche often criticized English philosophers and praised French philosophers. Could he read English and French? Or did he read translations of English and French philosophical writings?

According to the Nietzsche Channel, Nietzsche read Dostoevsky's Notes from Underground in a French translation by E. Halpérine et Ch. Morice.

The Nietzsche Library at the Nietzsche Channel contains a reconstruction of his library. This collection lists plays by Shakespeare in English along with translations in German. The works of Shelley were in German translation. He had a copy of Francis Galton's Inquiry into human faculty and development.

From the above, I assume he could read French. Searching this reconstruction of his library may provide suggestions about the extent he was familiar with either French or English.

Reference

The Nietzsche Channel http://www.thenietzschechannel.com/

Frank Hubery has answered the question for Nietzsche's knowledge of French. I'd add the following extract from Nietzsche's April 1875 letter to Marie Baumgartner :

He himself confirms the shakiness of his French in a letter to Marie
Baumgartner of April 1875, when he thanks her for correcting his mistranslation
from Montaigne in Schopenhauer as Educator: “To the discoverer of my mistake
many thanks; you see my French is in rather a sorry state, and before I idealize
Montaigne I should at least understand him properly”. (Duncan Large, 'Nietzsche and/in/on Translation', Journal of Nietzsche Studies , Vol. 43, No. 1, Special Issue: Proceedings from the North American Nietzsche Society (Spring 2012), pp. 57-67 : 59.) If Nietzsche's French was in a 'sorry state' he must have had an elementary acquaintance with the language.

One might note Brobjer's comment :

On the
whole, Nietzsche's French was such that he did not read fiction in French before about 1880. ((Thomas H. Brobjer, 'Nietzsche's Reading and Private Library, 1885-1889', Journal of the History of Ideas, Vol. 58, No. 4 (Oct., 1997), pp. 663-693 : 666.)

As to English the following extract is one piece of evidence :

Towards a Genealogy of Morals (1887) is to a large extent a response to Nietzsche's reading of and about "English psychologists," including Rée; and in the history of law (his library contains ten books within this field
and the majority of these are annotated by Nietzsche). (Thomas H. Brobjer: 673.)

'...reading of "English psychologists"' points tentatively to some knowledge of English but this quote suffers from the rather serious defect that Paul Rée (1849–1901) was a German, not an English, psychologist.

English words do occur in Nietzsche's writing, of which Duncan Large observes that :

... the linguistic diversity is at times dazzlingly
extreme. Foreign words abound—mostly Latin and French but also ancient
Greek, Italian, English, Spanish, Provençal, Sanskrit. (Duncan Large, 'Nietzsche and/in/on Translation', Journal of Nietzsche Studies , Vol. 43, No. 1, Special Issue: Proceedings from the North American Nietzsche Society (Spring 2012), pp. 57-67 : 60.)

However, knowledge of individual words does not demonstrate any considerable knowledge of the language.

Overall I accept the judgement of Duncan Large :

As an academic classicist
Nietzsche himself translated between German, Greek, and Latin, but his command
of modern foreign languages was relatively unimpressive, and he viewed
language learning as a necessary evil, looking forward to the time when a new
lingua franca would obviate the need for language learning or translation at all. (Large: 57.)

If Nietzsche did tackle English it is likely to have been with the help of a dictionary. There is, so far as I know, no indication that he read or wrote the language with command or fluency.