qt: Poll ShutdownTimer after init is done

Summary:
Backport of Core PR12377
https://github.com/bitcoin/bitcoin/pull/12377

Test Plan: ninja check check-functional

Reviewers: deadalnix, Fabien, #bitcoin_abc, markblundeberg

Reviewed By: #bitcoin_abc, markblundeberg

Subscribers: teamcity, schancel

Differential Revision: https://reviews.bitcoinabc.org/D2545