NestJS is a backend framework and Next.js is a React framework, so the real choice is whether Next.js route handlers are enough for your API.
In our lab NestJS on Fastify served 45,863 requests per second, NestJS on Express 12,333 and a Next.js route handler 4,261 for the same validated endpoint.
Putting NestJS behind a Next.js page added about 13 ms at the median, so add it when a second client, background jobs or a larger team justify the hop.