Standard process during a disaster
"When disaster strikes" isn't a one-size-fits-all situation. The HADR worksheet from brentozar.com includes multiple categories, each with specific failure scenarios. Then, RPO &amp; RTO are defined for each of those failure scenarios. I'd recommend you use this worksheet, or make one similar to it. 

Next, you will want to develop a DR Plan for what you will do when those scenarios arise. DR Plans are organization-wide documents that involve not just data, but also detail failing over applications, infrastructure, etc. This is especially important for a failover to DR that involves data loss. You will want to pre-define who needs to be involved, what actions are needed for failover, and how long it takes to perform those actions. When data loss is involved, one of the most important actors in your DR Plan is identifying who will make the decision to accept data loss &amp; fail over to your DR site. This is usually a management or business decision.

In the event one of these scenarios happens (ex, a Storage outage), that worksheet and DR Plan will guide you on how long you work to resolve the failure situation in your primary data center before beginning failover to DR. 

Lets look at an example:

Let's say your RTO is 4 hours during a widespread outage. Your RPO is 20 minutes of data loss, and you are using 10-minute log shipping to your DR site, and this is working as designed. Your DR Plan estimates failover will take 90 minutes. The Director of IT and CTO are designated as being allowed to authorize failover to DR with data loss.

A complete power outage affects your primary data center, and all systems are completely offline. Because your log shipping is up to date, failing over would cause only 10 minutes of data loss--well within your 20-minute RPO goal. You can wait up to 2.5 hours for the power to come back online, and still meet your RTO goal. 

The CTO or Director of IT would be charged with monitoring the situation and making the decision if and when to fail over to DR. If they choose to fail over at or before the 2.5 hour mark, you should meet your RTO target. If they wait longer, you would miss your RTO target--that's ultimately a business decision, not a technical one. As a DBA, your job is to provide enough information about the failure (and recovery/failover process) so the designated person can make that decision.