I saw this last night and was hoping that some knowledgeable person would have composed an illuminating answer for me to read this morning. No such luck, but I'm sure some here can give a better answer than the following attempt, which I provide in the hope of eliciting one.
The Church of England and the Lutheran Churches are quite similar in a lot of respects, and are generally in communion with each other. The former however views itself as the national instance of the universal Church, and in that regard self-identifies as Catholic. It is reformed in certain of its doctrines and practices, but acknowledges the authority of tradition as opposed to the Lutheran dogma of sola scriptura, meaning that the only authority is Scripture and all we need to know of faith is therein.
Historically the Lutheran Churches are rooted in the teachings and example of Martin Luther, who is not nearly so directly influential in the Church of England. The latter has an image of being fuzzy and indistinct about doctrines and practices in general, which is correct; it was purpose-designed by Elizabeth I, who in many ways was its true founder, to be as broad a church as possible, her wish being to accommodate all her people within it. The various Lutheran Churches were not so driven, and are more distinctly Reformed in nature, though they remain liturgical and episcopal. This along with specific points of doctrine distinguishes them as well as the Church of England from the evangelical churches, which would include the various denominations deriving from Calvinism.
The best-known Calvinist doctrine is predestination, which the Lutherans share but have a more subtle and ambiguous view of, and the Church of England accommodates, as it does a wide variety of other views, but does not teach. To Calvinists, God's redemptive grace is only for the elect; Christ came to save a chosen part of mankind, and those who are saved, those who believe in and practice Christianity, do so not of free will but because God elected them to so believe and practice. To Lutherans, God's redemptive grace is for all mankind but only those elected by God partake of it. It's not a difference that makes no difference, the division between Christ coming to save everyone and Christ coming to save the chosen is quite fundamental, but it must be admitted that the practical outcome is the same.
The Church of England does not as I understand it have a specific doctrine on the question. While it certainly has doctrines it is a church of externals and participation, which can be carried too far of course but is an approach with many merits also. A reputed off-the-cuff remark of Elizabeth's late in life is perhaps the key to it: "There is one faith, one Christ ... all else is but disputes over trifles."
To summarise: of the three, the Church of England considers itself the reformed universal Church, English branch. The Lutheran Churches consider themselves the reformed Christian religion. Churches following the Calvinist tradition consider themselves the gatherings of God's Elect, which is meant to be not a put-down of them but descriptive. I am sure someone can do a much better job and put me right on various points, but for what this is worth I offer it.
PS I hadn't seen Drake and Uriel's answers when I started the above. I agree entirely with Drake, and would add an anecdote to Uriel's; an Italian friend who can be a bit vague about things was visiting London and decided to go to Mass, electing St Paul's for the purpose. He was quite comfortable until the celebrant being a woman clued him in that he might have made a slight error...