Accessibility isn't just a technical issue. It's a design issue. It's a content issue. It's something that every department that touches a website should think about.
Accessibility is about people. Not people with disabilities.
I see an audit as a way to correct issues prior to a website going live.
Due to time constraints, I almost always try and build in accessibility and usability from the off. That way, an important part of how a website should be built isn't left out.
At the end of a project I check the website for glaring issues from a build point of view. I prefer to do this once a client has populated everything as how they populate can also affect accessibility.
Google Chromes Accessibility Developer Tools is brilliant for spotting these glaring issues. I usually run an accessibility audit from console and fix any issues that arise.
Testing a website with real disabled users is the only way to ensure it offers optimum accessibility but I realise that this option isn't always available.
I check for colour contrast issues using Webaims checker before anything else. Usually I can eyeball if something is bad and try to aim for WCAG AA.
If accessibility fails badly to the point that even I myself am struggling, I will pass these issues back to the designers for a rework.
I personally rarely use a mouse. I have one at work but prefer the keyboard. At home I don't use a mouse. Keyboard testing for the most part is easy. Developers tend to develop for themselves which usually means a lack of visual feedback upon tabbing through a website. My approach to testing is usually just using my keyboard to navigate around the website. It's as simple as that.
Screen Readers & Assistive Technologies
Testing content with a screen reader is made easy by using Chrome Vox. Anything that doesn't make sense verbally, gets fixed.