When Multiple nouns have single ownership the possessive form is applicable to the second noun.
David and John's adventures are hilarious.
In the sentence choosen by the OP,
one is a noun and the other is pronoun in a possessive case.
We can not say paul and mine adventures.
So the correct option seems to be:
Paul and my trip to London
Here we can not say Paul trip to London or My trip to London.. They are going together. So it is correct say Paul and my trip rather than Paul's and my trip to London.
We can not say Jack's and jill's adventures.
Here is a link which explains the rules: