chore(nextjs): Improve experience when swapping keys on Keyless mode #11179

Job	Run time
check-permissions	2s
Unit Tests (18)	41s
Formatting, linting & changeset checks	1m 33s
Unit Tests (22)	39s
Integration Tests (generic, chrome)	1m 24s
Integration Tests (express, chrome)	1m 30s
Integration Tests (quickstart, chrome)	2m 6s
Integration Tests (ap-flows, chrome)	2m 31s
Integration Tests (elements, chrome)	2m 40s
Integration Tests (sessions, chrome)	3m 34s
Integration Tests (astro, chrome)	1m 30s
Integration Tests (expo-web, chrome)	1m 30s
Integration Tests (tanstack-start, chrome)	1m 27s
Integration Tests (tanstack-router, chrome)	1m 23s
Integration Tests (vue, chrome)	1m 31s
Integration Tests (nuxt, chrome)	1m 29s
Integration Tests (react-router, chrome)	1m 27s
Integration Tests (nextjs, chrome, 13)	5m 41s
Integration Tests (nextjs, chrome, 14)	5m 54s
Integration Tests (nextjs, chrome, 15)	5m 48s
	44m 20s

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