“Forgotten” isn’t really the right word, since Old English is English, which exists as a continuum from about the fifth century to today. So really the question should be was there a time in which people no longer had the ability to read English from the Old English period, and if so, who figured out how to read it again? It may not seem like an important distinction but it is. The Old English period last for about 600 years and so it's not right to think of Old English as a single, monolithic entity that suddenly ceased to exist and then was forgotten.

But to answer the question, yes, the ability to read Old English was slowly forgotten. People in the twelfth century who had need to read older records (church records, laws, sermons, etc.) still could, and probably without a great deal of difficulty. A lot of texts copied in English of this period are somewhat updated to reflect newer spelling usage, like using the letter v for f, or perhaps reflect a breakdown in older spelling conventions, but many of the older forms are retained. For example, in the Peterborough Chronicle entries from the 1150s, they still use the letters þ, ð, and æ, sc for the “sh” sound, c for both the “k” and “ch” sounds, etc., just like in Old English, but a century later, only the letter þ was still used. ð and æ had dropped out of usage, the “sh” sound was written in a variety of ways (such as sh, sch, s), and so on.

In the thirteenth century, the copying of manuscripts in Old English diminished. Perhaps the most famous reader of Old English in this period is known as the “Tremulous Hand of Worcester,” so called because his handwriting is very shaky. He wrote translations of difficult Old English words in the margins of numerous manuscripts, translating them into Latin or into Middle English forms. In the past it was often suggested that he must have been very old, and thus remembered earlier forms of the language more than younger readers, but now it is thought that he had a neurological condition. You can read about him in the excellent book by Christine Franzen. The need he felt to write translations of Old English suggests that few people in his day could read Old English and he is often considered one of the last people able to understand it.

From him we jump up to the early sixteenth century -- the rediscovery of Old English. This was pioneered by scholars such as John Joscelyn (born 1529) and Lawrence Nowell, among many others. Joscelyn worked for Matthew Parker, Archbishop of Canterbury under Queen Elizabeth, creating a dictionary of Old English and producing editions of Old English texts. The first to appear was their publication in 1566 of A Testimonie of Antiquitie, which contained editions of the Easter Homilies of Ælfric, among other texts, which were used as evidence for the early independence of the English church from Rome. This was the main goal of the 16th-century readers of Old English, and they ransacked monastic libraries for manuscripts (helped greatly by Henry VIII’s dissolution of the monasteries in the late 1530s). Parker gathered up a huge collection of manuscripts (now held in the library of Corpus Christi College in Cambridge) which he and Joscelyn went through for relevant texts and from which Joscelyn worked to create a grammar and a dictionary. Nowell, who owned the manuscript which contained Beowulf, also worked on a dictionary of Old English. None of these dictionaries were published but they circulated in manuscript form and were used to help produce a steady stream of publications of Old English texts in the latter half of the sixteenth century, into the seventeenth. Though Joscelyn did make a grammar of Old English, it remained unpublished like his dictionary, and subsequently went missing around 1612 when Robert Cotton lent it to William Camden (Cotton owned the largest collection of OE manuscripts, most of which are now in the British Library). In 1659 Ælfric’s Grammar and Glossary was finally published. This is a grammar and glossary of Latin, but written in Old English and so it proved incredibly useful for others who wanted to learn to read. Other grammars of Old English were published in the 17th century, written in Latin, which all educated people then could read, and then in 1715 Elizabeth Elstob published a grammar of Old English written in English.

To sum up, most of the work recovering Old English was done by churchmen and antiquaries in the 16th century who were primarily interested in using the texts to support the new Protestant religion and demonstrate that the English Church from its earliest days had been independent from Rome. They produced dictionaries and grammars, though these only circulated in manuscript form within a small circle of scholars. In the 17th and early 18th century grammars and dictionaries started to appear in published form, allowing the tools necessary to learn Old English to reach a wider audience, and more and more works started appearing. It wasn’t until the 19th-century though that literary scholars turned to Old English poetic texts like Beowulf but even then much of the publication of Old English texts in the 19th century came about through the Early English Text Society, founded to produce published editions of early texts for use in creating the Oxford English Dictionary.

More information can be found in T. Graham’s The Recovery of Old English.