Per the FAQ on Keane's website, Keane's drummer Richard Hughes, stated the following:
"We've been asked whether "Somewhere Only We Know" is about a specific place, and Tim has been saying that, for him, or us as individuals, it might be about a geographical space, or a feeling; it can mean something individual to each person, and they can interpret it to a memory of theirs... It's perhaps more of a theme rather than a specific message... Feelings that may be universal, without necessarily being totally specific to us, or a place, or a time..."
With the nostalgic sentiment and the overall tone of the song, I think Keane is attempting to express a Portuguese term known as 'saudade', which does not have a direct English translation but roughly means "that which we remember because it is gone."
Babooshka
Babooshka
She wanted to test her husband
She knew exactly what to do
A pseudonym to fool him
She couldn't have made a worse move
She sent him scented letters
And he received them with a strange delight
Just like his wife
But how she was before the tears
And how she was before the years flew by
And how she was when she was beautiful
She signed the letter
All yours
Babooshka, Babooshka, Babooshka-ya-ya
All yours
Babooshka, Babooshka, Babooshka-ya-ya
Babooshka
She wanted to take it further
So she arranged a place to go
To see if he
Would fall for her incognito
And when he laid eyes on her
He got the feeling they had met before
Uncanny how she
Reminds him of his little lady
Capacity to give him all he needs
Just like his wife before she freezed on him
Just like his wife when she was beautiful
He shouted out
I'm all yours
Babooshka, Babooshka, Babooshka-ya-ya
All yours
Babooshka, Babooshka, Babooshka-ya-ya
All yours
Babooshka, Babooshka, Babooshka-ya-ya
Babooshka, Babooshka, Babooshka-ya-ya
Babooshka, Babooshka, Babooshka-ya-ya
Babooshka, Babooshka, Babooshka-ya-ya
Babooshka-ya-ya
Babooshka
Babooshka
Babooshka
Babooshka
She wanted to test her husband
She knew exactly what to do
A pseudonym to fool him
She couldn't have made a worse move
She sent him scented letters
And he received them with a strange delight
Just like his wife
But how she was before the tears
And how she was before the years flew by
And how she was when she was beautiful
She signed the letter
All yours
Babooshka, Babooshka, Babooshka-ya-ya
All yours
Babooshka, Babooshka, Babooshka-ya-ya
Babooshka
She wanted to take it further
So she arranged a place to go
To see if he
Would fall for her incognito
And when he laid eyes on her
He got the feeling they had met before
Uncanny how she
Reminds him of his little lady
Capacity to give him all he needs
Just like his wife before she freezed on him
Just like his wife when she was beautiful
He shouted out
I'm all yours
Babooshka, Babooshka, Babooshka-ya-ya
All yours
Babooshka, Babooshka, Babooshka-ya-ya
All yours
Babooshka, Babooshka, Babooshka-ya-ya
Babooshka, Babooshka, Babooshka-ya-ya
Babooshka, Babooshka, Babooshka-ya-ya
Babooshka, Babooshka, Babooshka-ya-ya
Babooshka-ya-ya
Babooshka
Babooshka
Babooshka
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Somewhere Only We Know
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Since the song ends with him telling Babooshka he's all hers, you don't get to hear what came after when he finds out it's really his wife. But I like to think that such an encounter has a positive ending, along the lines of, "Hey, it's actually you all along! What happened to our live together? Why can't you be this way as yourself?"
If she couldn't, then the relationship is over. If she can, well, it opens up a new chapter in their lives.
So for a long time the song had a happy end for me. He still loves his wife, but longs for a more playful relationship.