You are confusing language, culture, religion, law and the political system and intellectual structure and rolling them up into one big ball. The latter two were drawn from European thought - not only British, but Continental. The legal system was based on the British common law system, although with modifications (except for Louisiana, which has a system based on Roman law, among other things).
Religion can be anything within the Judeo Christian sphere; Maryland was founded by Catholics and had a much more tolerant system than any of the other colonies that eventually became states. In addition, many low-church Protestants did not consider even the Anglicans to be Christians, so there was division even in what you are loosely calling the "Protestant" world. Congregationalism, which in our day is so liberal they are all just barely deists, was very rigid, authoritarian and theocratic, and our intellectual authors, such as Jefferson and others, were not part of it.
As for language, while the language of the US came very close to being German at one point, as the US consolidated and added states in the 19th century, the English language was always seen by any immigrant group as the way to get ahead and has always been learned by them. This doesn't mean they become English, though. Where would we be without such folks as the Eastern European Jewish immigrant kids who were the song writers of the 30s? But they didn't become WASPs - they simply wrote their songs in English, and wrote in a way that would appeal to the great majority of their fellow American citizens (who were by no means all WASPs).
I used to teach ESL for adults in New York and we had to turn people away from the classes. Some groups, such as Poles (the largest single group of illegals in NYC was for some time - and may still be - Poles, btw), lived very much in their own communities and would take a few classes and never return. But others, particularly Hispanics, were determined to get ahead and move beyond being hotel maids. So the secret with English is to make it more available, on the one hand, and more mandatory, on the other - no more "bilingual" ed (which means no education, actually), no more ballots in other languages, etc. The encouragement of the learning of English was actually part of Bush's immigration plan, which nobody here at FR liked and which, as a result, is being replaced by one that is much more "liberal."
As for culture, again, if the left had not been so aggressive at driving our shared holidays and traditions out of our schools, we would be much more integrated. But don't blame the immigrants. Put that energy into trying to get the schools to change their curricula and dump the five chapters on the glories of the Mayan calendar and instead put in a little bit about actual American history. It may interest you to know that North American Christmas customs are spreading throughout Latin America; at the same time, I don't see anything wrong with Americans taking some of the Latin American traditions, such as the Posadas, or the custom of building little replicas of Bethlehem or of bringing out the Niño for adoration after Christmas Mass. Many of the things that you probably think of as English in any case are actually German, and probably German Catholic, at that, since the Congregationalists and many other groups did not approve of religious imagery or celebrations and even Irish Catholics, because of centuries of persecution in their homeland, were a little weak on the colorful customs.
In other words, you are cutting your divisions too fine when you attempt to decide on what is a common culture. We already have a common culture that is not WASP culture. However, it is all essentially European culture, and the ideas that are behind it are European ideas, and the religion that shaped it is a Christianity that transmitted both the religion of the Jews and the philosophical understandings of Greece and Rome, shaped by the egalitarian revolution of Jesus Christ, where all were one and equal in Him.
The only question to me is whether we could survive without this, that is, under Islam, and my answer is no. For one thing, Islam is a theocracy (which is fundamentally anti-Christian and which even the Congregationalists finally rejected) and it has a legal system which has nothing to do with either common law or Roman law. In addition, it rejects the principle of equality under the law (which even the US has had to fight for) and in fact rejects the value of reason in any aspect of life. Not to mention the fact that it rejects figurative art, most music and dance, and just about anything we like in this society.
I'm all in favor of more Mexican immigrants, more Polish immigrants, more immigrants from any place that shares our essentially European philosophical background or is willing to respect it. Asian immigrants - who do not share it - respect it and have adapted very well. Muslims do not share it and do not respect it, and even those who have been born here and brought up in it despise it. So I think we have to cease worrying about things that are non-essential and have in fact already changed considerably over the centuries and have even varied from place to place and decide what is really essential. And then get the powers that be, such as the schools, to accept this.
While people here are sitting around hating Catholics and Hispanics, the Muslims are getting "prayer rooms" in the public schools and having the "call to prayer" broadcast on streets where the ACLU has tried to stop church bells.
I would add, don't blame the immigrants for the welfare system, the free medical care, the repeat felons back out on the streets, or the general failure to promote literacy and moral character. All of these outrages are perpetrated by the United States governments (federal, state, and local), whose officials are elected by the United States' citizens.
Illegal immigrants didn't elect the pack of thieves in Raleigh. (Maybe Yankees did ...)
All those things we try to separate in our mind. But in the actual reality they are entangled in one big ball.
The latter two were drawn from European thought - not only British, but Continental
"Drawing from [someone else] thought" usually not just replace what already is there. Rather some type of combination takes place underneath, with the deceiving appearance of likeness. When Japanese put on Western suits and ties, adopt Western institutions, you might think that they became same as West, just because they look the same.