Cranmer and other liturgists of the newly independent Church of England (but mostly Cranmer) took the Sarum Use of the Roman Rite, cut out what they believed to be theologically problematic, and reshaped it to reflect the doctrine of justification by faith alone and to make it clear that the Eucharist was not a sacrifice.
There were two stages to this: 1549 was much more cautious and much more compatible with Catholic theology. 1552 was much more clearly Protestant. 1552 could not, I think, be used by Catholics with a clear conscience; 1549 could. Scholars argue over whether Cranmer’s views genuinely changed between 1549 and 1552 or whether he always intended to lead people on gradually.
Under Elizabeth, the 1559 BCP was adopted, which basically followed 1552 but reintroduced some of the more conservative elements from 1549. The BCP presently in official use in England (though most parishes use contemporary alternatives from the Book of Common Worship, or in some low-church cases don’t use a fixed liturgy at all) is 1662, which is a slightly more conservative version that still basically follows the structure of 1552 (the main innovation being moving any language about sacrifice and offering oneself to God to the post-communion thanksgiving, separating it from the Eucharistic prayer itself).
The Scottish BCP drawn up in 1637 reflected a much more “high-church” theology and reunited the Eucharistic prayer, with language about a sacrifice of praise and thanksgiving being used before Communion. This was one reason why many Scottish Protestants rejected the BCP, sometimes violently. However, it was this BCP that was used in the late 18th century as the model for the American BCP. So the Episcopal Church inherited a liturgy that was more “Catholic” than the English tradition, particularly in the key point of having sacrifice language as part of the Eucharistic prayer itself.
The American Episcopal Church today officially uses the 1979 BCP, which includes a revision of the traditional Eucharistic prayer (two revisions, actually) as “Rite I” and more “Vatican-II-like” prayers as “Rite II.”
Edwin