This is an example of the present progressive tense being used to describe a temporary situation. In my opinion, the likeliest interpretation is that the speaker is referring to Tom's current living arrangements, so the month of time is likely already underway.
While not necessary or obvious from context, it's possible that we're closer to the beginning of that period too, otherwise I would've expected something like "Tom is living in London for another few weeks". Again, that can only be confirmed in context.
To Maulik's point, the first interpretation you provide is not supported by the example sentence. If Tom was indeed entering his second month in London, then we'd say:
Tom has been living in London for a month.
The use of a continuous tense here again hints that this is a temporary situation (which continues in the present). If we only wanted to focus on the fact that a month has elapsed, and we don't care whether Tom is still living in London, we could also say:
Tom has lived in London for a month.
As you can see, there is a subtle difference between the last two examples, so which tense you use would be determined by your intent.