If the email account wasn't password protected and you didn't hack into it, you've haven't committed an illegal act in reading mail that wasn't sent direct to you.
It's highly unlikely a prosecution would ensue, any more than opening your partner's postal mail will result in a prosecution, but it's worth noting that the offence in the Computer Misuse Act has nothing to do with passwords and hacking.
An S.1(1) offence is made out if:
(1)A person is guilty of an offence if—
(a)he causes a computer to perform any function with intent to secure access to any program or data held in any computer [F1, or to enable any such access to be secured] ;
(b)the access he intends to secure [F2, or to enable to be secured,] is unauthorised; and
(c)he knows at the time when he causes the computer to perform the function that that is the case.
Reading someone else's email, even if you're sat in front of a computer they have left logged in, a least arguably meets that test. (a) is the act of clicking on links. (b) would be the stuff that gets argued about in court: does the act of living in a relationship provide implied consent, and does leaving devices lying around unprotected within a jointly-occupied house grant that authorisation? (c) would follow from (b).
The reason I pick this up is that if, say, you walked into your boss's office, noticed that her machine was logged in and flicked through her email, you would not only be inviting a bunch of contract and employment law issues as soon as you started reading, but that would be manifestly a CMA S.1(1) offence the moment you did anything which altered what was on the screen. The offence is in instructing the computer to do something you aren't authorised to do, and argument will be about whether you were authorised, not how you got access.
Complex analogies involving doors, welcome mats, keys and locks, trespass and theft, beloved of people wanting to complicate computer law, aren't necessary: CMA stands on its own, and all that's necessary to start a prosecution is to show that the acts meet the S.1(1) test.
As I say, the police and CPS are hardly going to be interested. But be careful once you get outside your house...