to catch (a) cold/there wasn’t being any train for London at that time
1.
I don’t often get sick, but now and then I do catch a cold.
I don’t often get sick, but now and then I do catch cold.
-Between the two sentences, which is correct?

2.
She replied, “London,” but I didn’t think she was speaking the truth because there wasn’t any train for London at that time.
-This is the original sentence above, but can I say "there wasn’t being any train for London at that time."?

Question

to catch (a) cold/there wasn’t being any train for London at that time
1.
I don’t often get sick, but now and then I do catch a cold.
I don’t often get sick, but now and then I do catch cold.
-Between the two sentences, which is correct?

2.
She replied, “London,” but I didn’t think she was speaking the truth because there wasn’t any train for London at that time.
-This is the original sentence above, but can I say "there wasn’t being any train for London at that time."?

Answer

This!.⬇ 
... you'll catch cold!
​
Idiomatic, you will feel your body temperature drop (or feel cold).
... you'll catch a cold!
​
Not idiomatic, you will contract a case of Acute viral nasopharyngitis, or acute coryza, also known as the common cold.

So it this case you have to say
I don’t often get sick, but now and then I do catch a cold.

Answer

1) I don't know the formal answer, but both forms are used in spoken English. Personally, I prefer saying, "to catch a cold."

2) You would not say "there wasn't being."  It's a great question, but you don't add a present participle after "there is" or "there was."  You could say, "I/he/she was being," but never, "There was being."

Answer

1. I think to catch "a cold" is more commonly said in the U.S. So that is what I would say.  However, I think you could use either form.  I have heard both.
  
2. This is the original sentence above, but can I say "there wasn’t being any train for London at that time."?

No, adding "being" to the sentence is incorrect. You could also say "There was no train to London at that time."