What is the meaning of “cop” in: “if London cops it, he'll cop it”?

In British English 'cop it' can mean a few things but one of two of those things is certainly being referred to. From http://www.oxforddictionaries.com/definition/english/cop:

1.2 (cop it) British Get into trouble:
will you cop it from your dad if you get back late?
1.3 (cop it) British Be killed:
he almost copped it in a horrific accident

London is the city. The story is about nuclear war and Ron is referring to London being hit by a thermonuclear bomb. Both definitions would apply to that but I think the 'be killed' definition is the most likely. London would be annihilated and that could be thought of as being killed. Ron could hardly 'get' the blast without being killed. Calling being hit by a nuclear explosion as 'getting into trouble' would be to ridiculously understate it (he is British though, so that is a possibility).

That the immediate conclusion to the statement is that Ron is not going to build a shelter confirms this: Ron has said he will die if London dies.

The verb to cop has quite a few meanings, the most basic of which is ‘seize, grab’. The sense relevant here is definition 1.1 (of the verb) as given in the Oxford Dictionaries Online article:

Incur (something unwelcome)

If you cop something, it is forced on to you, you get it without wanting it. You can cop the blame for something; you can cop a bad deal from someone (meaning that you’re treated unfairly by them); you can cop one (‘one’ referring here to a punch) on the side of the head, etc.

Idiomatically, when you cop it (definitions 1.2 and 1.3), the ‘it’ in question can either be trouble in general, or more specifically death, which seems to be the sense intended here. So when you cop it, you incur death—in other words, you’re killed.

It’s a very colloquial, perhaps even slangy, expression, but it’s common enough in British English. In the script excerpt he, Ron is saying he accepts that if London gets killed (= blown to smithereens) in a nuclear blast, then he will get killed too.