Backport #59662 to 24.1: Fix query start time on non initial queries

[robot-ch-test-poll2]

Original pull-request #59662
Cherry-pick pull-request #59841

This pull-request is a last step of an automated backporting.
Treat it as a standard pull-request: look at the checks and resolve conflicts.
Merge it only if you intend to backport changes to the target branch, otherwise just close it.

[robot-ch-test-poll]

This is an automated comment for commit 5d928d3 with description of existing statuses. It's updated for the latest CI running

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Successful checks

Check name	Description	Status
CI running	A meta-check that indicates the running CI. Normally, it's in success or pending state. The failed status indicates some problems with the PR	✅ success
ClickHouse build check	Builds ClickHouse in various configurations for use in further steps. You have to fix the builds that fail. Build logs often has enough information to fix the error, but you might have to reproduce the failure locally. The cmake options can be found in the build log, grepping for cmake. Use these options and follow the general build process	✅ success
Compatibility check	Checks that clickhouse binary runs on distributions with old libc versions. If it fails, ask a maintainer for help	✅ success
Docker server and keeper images	There's no description for the check yet, please add it to tests/ci/ci_config.py:CHECK_DESCRIPTIONS	✅ success
Install packages	Checks that the built packages are installable in a clear environment	✅ success
Integration tests	The integration tests report. In parenthesis the package type is given, and in square brackets are the optional part/total tests	✅ success
Stateful tests	Runs stateful functional tests for ClickHouse binaries built in various configurations -- release, debug, with sanitizers, etc	✅ success
Stateless tests	Runs stateless functional tests for ClickHouse binaries built in various configurations -- release, debug, with sanitizers, etc	✅ success
Stress test	Runs stateless functional tests concurrently from several clients to detect concurrency-related errors	✅ success