"Victoria Tube line part shut hit by wet concrete flood" . Is this correct English?

No, because part is already an adverb meaning partly. According to the OED, this adverb can be used to modify a phrase, a verb, an adjective, or a participle.

For example, here’s a Shakespeare citation they give of this use:

• 1600 Shakespeare Henry IV, Pt. 2 i. iii. 60
  On..who (halfe thorough) Giues o're, and leaues his part-created cost, A naked subiect to the weeping clowdes.

And here’s a recent citation:

• 2000 N.Y. Times 30 Dec. a1/3
  Frankfurt..is comfortable being a part-Muslim city with 27 discreet mosques.

Since shut is an adjective (or a past participle), then part-shut (with or without the hyphen) is a condition meaning partially shut.

I know that the answer that argues that part is an adverb has been accepted already, but I do believe that another analysis is more likely here.

As the OP stated, if you actually go to the BBC site, you will notice that the caption of the photo reads,

[1] Part of the Victoria London Underground line has been suspended after wet concrete flooded a control room.

Here part is unambiguously a noun. (Note also the absence of a determiner in front of it, despite it being a countable noun in the singular; I'll return to that below.)

Once one notices that, one also notices that even in the headline itself, part can also be interpreted as a noun, part of the nominal Victoria tube line part, meaning 'a part of the Victoria tube line'.

Under this interpretation, the headline would, as a full sentence in Standard English, read something like this:

[2] A part of the Victoria tube line is shut after getting hit by a wet concrete flood.

A further note on grammar of the headline

Note that headlines have their own peculiar gramamtical rules, 'special styles of abbreviation', as CGEL calls them (p. 3). One well-known aspect of this is the dropping of some of the articles/determiners, even in front of countable nominals in the singular. CGEL gives the example

[3] Man bites dog, arrested.

Moreover, note the absence of a determiner in the figure caption [1], where part is unambiguously a noun.

Note also that there are other peculiarities with the title we are discussing. For example, we would normally expect something to appear between shut and hit—at least a comma, as would be standard in headlines (like in [3]).

Which interpretation is more likely?

So, provided we insert a comma (or something) between shut and hit, we have two possibilities: 1. part as an adverb modifying shut, or 2. part as a noun, the head of the NP (a) Victoria Tube line part (again headlines commonly drop certain articles).

If one looks at the OED entry for part as an adverb, it has three subentries: modifying a phrase (e.g. the ability to pay p͟a͟r͟t͟ in cash and p͟a͟r͟t͟ in green dollars for goods or services); modifying a verb (he p͟a͟r͟t͟-owns a nightclub out in Essex); and modifying an adjective or participle (e.g. Frankfurt..is comfortable being a p͟a͟r͟t͟-Muslim city with 27 discreet mosques.).

Another thing one notices is that in all of the examples given in OED for the last two—modifying a verb or modifying an adjective or participle—part is hyphenated.

Therefore, arguably, under interpretation of part as an adverb, the headline has a second typo (in addition to the absence of the comma between shut and hit), namely, the absence of a hyphen between part and shut.

But under the interpretation of part as a noun, the headline only has the one typo, the absence of a comma. As I have been stressing, the absence of a determiner in front of a countable nominal in the singular is not unusual in headlines, and so is arguably not a typo.

Therefore, the interpretation of part as a noun needs to postulate fewer typos than the interpretation of it as an adverb, and this is why I say it is preferred.