chore(nextjs): Improve experience when swapping keys on Keyless mode #11169

Job	Run time
check-permissions	2s
Unit Tests (18)	1m 28s
Formatting, linting & changeset checks	2m 9s
Unit Tests (22)	2m 13s
Integration Tests (generic, chrome)	4m 47s
Integration Tests (express, chrome)	2m 32s
Integration Tests (quickstart, chrome)	2m 43s
Integration Tests (ap-flows, chrome)	3m 7s
Integration Tests (elements, chrome)	3m 10s
Integration Tests (sessions, chrome)	4m 7s
Integration Tests (astro, chrome)	3m 28s
Integration Tests (expo-web, chrome)	3m 3s
Integration Tests (tanstack-start, chrome)	4m 5s
Integration Tests (tanstack-router, chrome)	2m 32s
Integration Tests (vue, chrome)	2m 29s
Integration Tests (nuxt, chrome)	2m 52s
Integration Tests (react-router, chrome)	2m 29s
Integration Tests (nextjs, chrome, 13)	6m 18s
Integration Tests (nextjs, chrome, 14)	6m 36s
Integration Tests (nextjs, chrome, 15)	6m 25s
	1h 6m 35s

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