If you are not taking care of your mobile version of your WordPress website then you are basically happily to lose over 50% of your traffic to competitors.
Here are some interesting stats:
“Percentage of all global web pages served to mobile phones from 2009 to 2018. This statistic presents the share of mobile phone website traffic worldwide. In 2018, 52.2 percent of all website traffic worldwide was generated through mobile phones, up from 50.3 percent in the previous year.” Source: Statista.
If you have a really old website most likely your website is not mobile optimised and if it’s outsourced overseas they skip this process altogether sometimes. A professional who cares about their work will make sure every looks relative to the desktop version from image sizes to backgrounds. The tricky part about getting mobile responsiveness on your WordPress website is that their are so many devices out their which means you need to develop for each one for complex layouts to make sure it looks good for all visitors.
How to test your WordPress website mobile score in Google
You can do easily by accessing: https://search.google.com/test/mobile-friendly
It will give you report on how Google sees your website. Try it and see, it’s free.