[backport#15091] Qt: update header count regardless of update delay

Summary:
https://github.com/bitcoin/bitcoin/pull/15091/commits/7bb45e4b7a71e0ff0e2bbeba5367022758efa5d4

Depends on D6177

Partial backport of Core PR15091

Test Plan:

ninja
./src/qt/bitcoin-qt

In a node that isn't 100% synced, see that overlay describes it is receiving headers,
and then blocks correctly.

Reviewers: #bitcoin_abc, jasonbcox

Reviewed By: #bitcoin_abc, jasonbcox

Differential Revision: https://reviews.bitcoinabc.org/D6179