We were almost close to booking tickets to London.
It's perfectly fine.
Consider: I look forward to seeing you
Here "to" is not a marker of the infinitive verb, it is a preposition, so the verb that follows is in the gerund (-ing).
And to be "almost close to" means you were nearly doing or about to do something. The "almost" could be dropped entirely or substituted with very, which is used to emphasize a description.
We were very close to booking tickets to London.
Oxford Dictionaries provides a definition and pertinent examples
1.3 (close to) Very near to (being or doing something)
- ‘I had a low point towards the end of last season and that is why I was close to leaving the club, but now I am at my best ever level.’
- ‘The attempt nearly succeeded as he was close to being able to get his head through the hole.’
references: "To hear" or "to hearing"? and Why was “to + verb+ing” used in this case?