It seems I found a way:
DISCLAIMER: I have no idea of what kind of side effects this may cause. I have a saved game in turn 249 (online speed) that ends at turn 251 with no "just one more turn" enable.
Did the deletions below, loaded the game, loaded that savegame and was able to go beyond 250. Stopped playing at turn 255, as I will not continue that game until tomorrow.
Hope it solves it.
I removed all references to "DEFEAT_TIME" from "Victories.xml" in ...\Steam\steamapps\common\Sid Meier's Civilization VI\Base\Assets\Gameplay\Data
Made a safe copy of Victories.xml" and removed the following lines from the original:
[~Line 11] <Row Type="DEFEAT_TIME" Kind="KIND_DEFEAT" />
...
[~Line 25] <Row DefeatType="DEFEAT_TIME" Name="LOC_DEFEAT_TIME_NAME" Blurb="LOC_DEFEAT_TIME_TEXT" RequirementSetId="TIME_DEFEAT_REQUIREMENTS"/>
...
[this whole block around line 350]
<!-- Time Defeat -->
<!-- Conditions - Score Victory must not be enabled and the Max Turns must be reached.-->
<RequirementSets>
<Row>
<RequirementSetId>TIME_DEFEAT_REQUIREMENTS</RequirementSetId>
<RequirementSetType>REQUIREMENTSET_TEST_ALL</RequirementSetType>
</Row>
</RequirementSets>
<RequirementSetRequirements>
<Row>
<RequirementSetId>TIME_DEFEAT_REQUIREMENTS</RequirementSetId>
<RequirementId>TIME_DEFEAT_SCORE_DISABLED</RequirementId>
</Row>
<Row>
<RequirementSetId>TIME_DEFEAT_REQUIREMENTS</RequirementSetId>
<RequirementId>TIME_DEFEAT_MAX_TURN_REACHED</RequirementId>
</Row>
</RequirementSetRequirements>
<Requirements>
<Row>
<RequirementId>TIME_DEFEAT_SCORE_DISABLED</RequirementId>
<RequirementType>REQUIREMENT_GAME_VICTORY_ENABLED</RequirementType>
<Inverse>1</Inverse>
</Row>
<Row>
<RequirementId>TIME_DEFEAT_MAX_TURN_REACHED</RequirementId>
<RequirementType>REQUIREMENT_GAME_TURN_MAX_REACHED</RequirementType>
</Row>
</Requirements>
<RequirementArguments>
<Row>
<RequirementId>TIME_DEFEAT_SCORE_DISABLED</RequirementId>
<Name>VictoryType</Name>
<Value>VICTORY_SCORE</Value>
</Row>
</RequirementArguments>