*tl;dr: When the background check comes up, consider mentioning to HR that the dates of your employment may show that you departed from your most recent role in April. Avoid justifying your actions until you're asked to explain.* 

I'm not a recruiter nor do I have extensive experience with this. I'll just pitch my thoughts.

I think as long as you didn't lie to them during the interviews after you were impacted by restructuring, you should be fine.

Maybe if anything, when the background check comes up, send a note to the HR team and let them know about the restructuring. I wouldn't say anything about why you didn't bring it up earlier as there's a risk of it being interpreted as malicious intent to withhold information; only address it if they ask you why you didn't mention it sooner. If it comes up, say you didn't think it was relevant to your candidacy. As you said, they never asked so you don't really need to tell.

I'm assuming they didn't interview you and give you the offer on the condition that you are currently working. The interview process and offer has shown that they like you and what you bring to the table.

On principle, they shouldn't rescind your offer if they never asked and you never lied.

edit: 

1. Between email or phone call, either works. I just wouldn't schedule a phone call just to tell them. Only if you already have a phone call scheduled for a different reason.
2. Depending on where you are, some (most? all?) companies will only disclose your employment dates and nothing more, even if your layoff was performance related. Explain it as a layoff.