English rosaries after Henry VIII
After Henry VIII's break with Rome, the rosary became suspect as a "Popish practice." In particular, the prayers of the rosary were problematic because they invoke a saint, the Virgin Mary. Article 22 of the Thirty-Nine Articles (which define the Church of England) describes the invocation of saints as "a fond thing vainly invented and grounded upon no warranty of Scripture, but rather repugnant to the word of God."
People of a more Puritan way of thinking, of course, regarded crosses, rosaries, the sign of the cross, et cetera as "Popish idols" and universally condemned their use. We have stories of parents snatching their babies out of the priest's arms at baptism in order to prevent him making the sign of the cross on their foreheads!
In 1571, the year after Queen Elizabeth was formally excommunicated by the Pope, a statute was passed declaring rosaries and other Roman Catholic accessories illegal. We have a picture published by Bernard Garter of London in 1579 of "Certaine of the Popes merchandize lately sen[t] over into Englande" which includes two very recognizable five-decade rosaries, along with a rather odd circle of nine striped beads and several plain ones, several religious medals, an "Agnus Dei" (consecrated wax medallion), religious woodblock prints, a pouch of "granum benedictum," and a portable altar.
However there are indications that the rosary did survive, at least in areas where Roman Catholicism remained strong, such as the north of England and the southwest (Devon and Cornwall). Dr. Madeleine Gray wrote a couple of years ago that there's a record of a seventeenth-century Puritan clergyman in south-west Wales who was reduced to imploring his congregation to use their rosaries "prayerfully and with thought." I think the jury is still out on whether the rosary continued to be used by members of the reformed English church, although it's certainly possible that otherwise good members of the Reformed church retained some "superstitious" traditions.
A good deal of comment has been occasioned by the discovery in Jamestown (Virginia) of beads that look like they could be part of a rosary. There is an online article about these beads (and others) in The Journal of the Jamestown Rediscovery Center, a refereed online journal. (See especially section 2.2 and endnote 3.)
If you have the time and inclination, you can also download the 1997 Jamestown excavation report, which includes details of a crucifix found on the site which also might have been part of a rosary. It's a 2.3 megabyte PDF file (which requires Acrobat Reader).
See pages 22 and 23 of the file for a description of the crucifix with a photograph, as well as some discussion of a possible Catholic presence at Jamestown and the survivial of Catholic traditions in Elizabethan England.