"All doors will not open" does not necessarily mean the same as "not all doors will open." From a strictly logical standpoint, "not all doors will open" is stating that the phrase "all doors will open" is false. It could refer to only some doors opening or zero doors opening. Practically speaking it would be used when some doors open and some remain closed.
There are multiple cases to examine, and I'll consider how each might be said in this case. Note that there are dozens of ways of phrasing each of these and that the London Underground probably has its own particular phrasing.
Every door on the train opens.
This would likely be said as, "All doors will open."
Zero doors on the train open.
This could be said in multiple ways. "No doors will open" or "All doors will remain closed" would be fairly reasonable.
Some doors open, but some remain closed.
This again could be said in multiple ways. "Not all doors will open," "Some doors will open," or "Some doors will not open." There are a bunch of other ways of constructing a sentence with this same meaning.
If the statement is that all of a category will (or will not) do X, then it is a statement about the properties of every member of that category. This would not be partial negation. In your example, "All doors will not open," means that zero doors will open. (Note that this is not idiomatically correct, but it would be understood.)
To rephrase the question you added in the edited portion, "Neither front door will open" or "The front two doors will not open" mean only that the front doors of the train will not open, but it implies that some other door on the train will, presumably those in the back.