Not in any kind of reliable way.
State governments are left to their own devices to determine how to conduct their elections. This means that the policies that govern those processes are subject to whatever institutional inertia normally exists for policy changes therein.
Add to this the fact that what states are battleground states is context driven, and not certain (Trump turned Wisconsin, Michigan, and Pennsylvania into wins for him, despite their status as part of the 'Blue Wall') the rules governing the count of electoral results don't necessarily have time to change in response to discovering they are a battleground state for this election.
The COVID-19 induced a swell of mail-in ballots that is illustrative here. Some states coped well with the (very predictable) surge of mail-in voting, allowing officials to begin processing and tabulating those ballots ahead of the election (though not releasing results until after the polls closed on election day). While others, including putative battleground states, did not make necessary changes - resulting in lengthy delays.
Certainly the workers of the polls feel the pressure, but the speed of the count is a function of choices made months - sometimes years - in advance and may or may not be reflective of the hindsight understanding of the state as a battleground.
Certainly some states will experience political pressure to change their policies in the manner you're asking about, but it's by no means certain to succeed and if it does so - again, it's on a state by state basis.
So while the statement about New York and California may be true (because a lack of pressure will certainly not cause anyone to hurry), the reverse is not necessarily true and demonstrably false in several recent instances.