[v25.2.x] [CORE-14007] Fix timeout handling in TS

[vbotbuildovich]

Backport #28081 to branch v25.2.x. This issue was created as a placeholder because the original PR's commit(s) could not be automatically cherry-picked.
Here are the commands to execute:

git checkout v25.2.x
git checkout -b manual-backport-28081-v25.2.x-92
git cherry-pick -x 441ee40c3a 6c38c1163c 89a527b0b4 4100c83423

git push origin manual-backport-28081-v25.2.x-92
gh pr create \
  --title "[v25.2.x] [CORE-14007] Fix timeout handling in TS" \
  --base "v25.2.x" \
  --label "kind/backport" \
  --head "manual-backport-28081-v25.2.x-92" \
  --draft \
  --repo "redpanda-data/redpanda" \
  --milestone "v25.2.x-next" \
  --body "Backport of PR https://github.com/redpanda-data/redpanda/pull/28081 "

[CORE-14007]: https://redpandadata.atlassian.net/browse/CORE-14007?atlOrigin=eyJpIjoiNWRkNTljNzYxNjVmNDY3MDlhMDU5Y2ZhYzA5YTRkZjUiLCJwIjoiZ2l0aHViLWNvbS1KU1cifQ