A reader has wondered why I left out that in the quote: England expects that every man will do his duty. Rather than take me to task for the omission, she has asked whether I had any ‘higher’ motive (stylistic?) in leaving out the subordinator.
I am flattered at the high opinion the reader has of me. But the plain fact is I had no higher motive.  I wrote the sentence as I found it in an American reference book. (I was, at the time, in Fremont, CA.)  I suspect the Americans leave out that in this case, while the British authorities regularly keep it.  Since I advocate (and use) British English, I would have definitely retained that if I had, at hand, a British reference book.  In any case a quotation should be given as it is in the original.  You can’t try to change or improve it—unless you are using it creatively. A little learning is a dangerous thing. That’s Pope. A knowledgeable young man may say: ‘A little flirting is a harmless thing.’
Having said that let me use the occasion to some good purpose.  Let us see in what cases the subordinator that can be (and usually is) left out.
That appears as clause introducer in several structures: 1. With clauses that report what somebody has said. The Minister said that he had been misquoted. 2. With clauses that express what one thinks, believes, etc. I believe (that) there has been some bungling. / I know (that) he can do it. 3. With clauses that explicate the content of a preceding noun (phrase), which is the ‘head’ of the clause. The view that the earth is getting warmer is gaining ground. The idea that science can explain everything is a modern myth.  4. In relative clauses: the paintings that you showed me. 5. And finally as part of a two-part conjunction: such that, so that, etc.
In the first structure, reported speech, that is often omitted in informal speech. I said (that) I was not aware of it. / I told you (that) he meant no harm. But with some verbs that can’t be dropped. He replied that it was not so. He pointed out that there was a mistake in the calculation. We noted that there was some excess payment. The verbs here are not just ‘reporting’ verbs. They are verbs, which indicate some mental activity on the part of the speakers making the observations.
Perhaps I should note separately the idiomatic phrase I’m afraid. This is never followed by that. I’m afraid it can’t be done.
(*I’m afraid that it can’t be done.)
With the second set of structures involving verbs like believe, think, imagine, etc. that is optional; in informal speech and writing, invariably dropped. One example should do. I think you are right.
In the third set of structures, where the clause explicates the content of the preceding noun (phrase) that has to be used and cannot be dropped. The view that temperatures are raising…. / The suggestion that school hours be reduced…. / The proposal that war be declared on illiteracy…. (Clauses like these are called ‘content’ clauses. The term is from Jespersen.)
As a relative pronoun that can be omitted (as pointed out sometime earlier in this column) when it stands for the object in the clause. The paintings that you showed me.  (Here that stands for the paintings.)  But in The messenger that came yesterday, that is the subject of the clause. The messenger came yesterday.  So it cannot be deleted.
As a conjunction that is found in structures like Parents work hard so (that) their children can have an easy time. / The speaker was such a bore (that) I quickly left the room. You can borrow my CDs provided (that) you use them carefully. In these cases that can be omitted.
To summarize: that cannot be dropped when it stands for the subject in relative clauses; (2.) when it introduces a content clause;   (3) with certain verbs which are not reporting verbs but indicate the state of the speaker’s mind.  In all other cases it can be dropped and, in informal speech and writing, is usually dropped.
Now to go back to Nelson’s message. Expect is not a reporting verb like say.  So it need not be followed by that.
England expects every man will do his duty is fine.  Why that?
Why indeed! According to some authorities this was the message flashed from Nelson’s ship.  Initially Nelson had England confides that…. But the signal officer pointed out that there were technical difficulties in flashing confide.  So it was changed to England expects that…. And this was again changed at the last minute to England expects every man will do his duty.
The account given above is taken from two sources, one British and the other American. The British source is Brewer: Dictionary of Phrase and Fable; The American source is: Robert Hendrickson: Word and Phrase Origins (2004). The Oxford Dictionary of Phrase and Fable doesn’t list the quotation at all.
It is amazing how on such a famous historical message like this one, there is no unanimity among the authorities as to what the actual message was.

Maxims and Observations of Kay S Wye
One can understand the jubilation among members of political parties when they win an election. They have now secured, legally, the keys to the state treasury.