I cannot say about the CofE but my experience in North America is that they will allow anyone to receive whether Anglican or not.If you went to an Orthodox Church the priest would not give you communion unless he knew you were Orthodox. It is suggested that you contact the priest to let him know that you will be visiting this way he is not surprised by you coming up to the chalice. If however, if you go up and he doesn't know you he will ask if you are Orthodox and if you are properly prepared to receive. If you are not he will allow you to kiss the chalice only.
Yes, the Anglican Church (as I said) will invite anyone who's a communicant member of their own Church to come and receive communion. The point is, they leave it up to you to be honest and act in good faith - it's not that they don't care, it's to do with being true to the spirit of Christ's sacrifice and his ministry, which didn't cling to the old Jewish laws restricting who could touch whom, and who could participate in what. That's the thinking as I've heard it explained, anyway.
You see, as far as I can tell, it'd be just as easy to lie to an Orthodox priest ('oh, yes, of course I'm Orthodox, I worship at St. Nicholas in Faraway Town, you must know it'), as it would be to lie to an Anglican one. The only difference is that, if you're in an Anglican Church, the understanding is that communicant members of all churches are automatically eligible to receive the Eucharist - that's not to do with not caring, but to do with our different attitudes towards community and the One True Church.
What I'm getting at is, there is a sincerity in the Anglican approach, which I think you are not seeing.