I've always thought that the Technical Writer should be part of the strategic roadmap definitions and be involved in the product inception ceremonies.
I keep the same opinion when talking about the UX Writer.
In terms of process, especially in Agile Design-driven Development, the first step of the UX Writing process, for me, is the Requirement Analysis.
"UX writing is the act of writing copy for user-facing touchpoints. This copy must not only embody the voice of the organization, but must also be considerate and useful for the user."
Understand the problem to be solved. Analyze use cases and be part of Design Sprint. Keep Personas in mind and write the first draft within the experience.
Write in Plain Language. Hemingway.app is a great resource for readability levels checking. Keep accessibility in mind. Choice of words is in agreement with Tone & Voice.
Evaluate technical accuracy of the content. Evaluate the emotional reaction to the content. Bring several options to the table. Experiment but keep the business process in mind.
User test prototypes and flow with UI microcopy. Validate requirements with PO, make sure microcopy is in compliance.
Complement requirements with prototypes displaying correct and final microcopy. Provide additional artifacts if needed for DEV.
Alpha testing prototypes + UI microcopy. Verify flows are consistent and that the final product reflects Design decisions.