If I was to go back in time to Medieval England (c 1000) and visit a rural hamlet, would I be able to have conversations with the locals? What about London?

You wouldn't be able to understand people in England during that era, modern English has received many words from Norman French, after the Norman conquest in 1066, before 1066 the language people in England spoke would have been a dialect of Old English, which isn't particularly intelligible to modern English.

It would become possible for a modern English speaker to understand a speaker in the 13th or 14th century with some thinking and guesswork. And it would be possible to understand a speaker from the 16th century with only a little difficulty.

The "modern" language that would be most useful for conversing with people in that era would be Latin. Since Latin was not a language used in every day conversation, therefore someone learning Latin both now and in the 11th century would be learning it from the same ancient texts.

To show how difficult Old English would be for a speaker of modern English to understand I'm going to quote a riddle from the Exeter Book, which was a book written sometime between 960-990.

Ic eom wunderlicu wiht wifum on hyhteneahbuendum nyt; nægum sceþþeburgsittendra nymthe bonan anum.Staþol min is steapheah stonde ic on beddeneoðan ruh nathwær. Neþeð hwilumful cyrtenu ceorles dohtormodwlonc meowle þæt heo on mec griperæseð mec on reodne reafath min heafodfegeð mec on fæsten. Feleþ sonamines gemotes seo þe mec nearwaðwif wundenlocc. Wæt bið þæt eage.