There's no need to optimize the phrase 'near me' in a keyword. Despite its frequent usage in searches, there are more effective approaches for optimization.
Google takes the 'near me' component from a user's search and pairs it with businesses and websites it deems are the most geographically closest and relevant to the user. So instead of 'Emergency plumber near me,' you would have 'Emergency plumber in London,' for example.
Therefore, for local businesses aiming for local rankings, including your specific target location within your keywords and tailoring your SEO strategy around them is advisable. In doing so, Google will understand the business's location and can recommend it to users who are searching for services 'near me.'