The casting time is 1 action. War Caster does not change it but bypasses it
The rules for spellcasting on page 202 PHB state:
Each spell description in chapter 11 begins with a block of information, including the spell’s name, level, school of magic, casting time, range, components, and duration. The rest of a spell entry describes the spell’s effect.
So the casting time of a spell is the time that is listed in the spell description.
The War Caster feat has this property:
When a hostile creature's movement provokes an opportunity attack from you, you can use your reaction to cast a spell at the creature, rather than making an opportunity attack. The spell must have a casting time of 1 action and must target only that creature
If War Caster changed the casting time of the spell to 1 reaction, it both could not use the spell (as the spell no longer would be a spell with a castig time of 1 action, that it must have), and it would need to say so. Compare the wording to the wording of the sorcerer's Quicken Spell feature that says:
When you cast a spell that has a casting time of 1 action, you can spend 2 sorcery points to change the casting time to 1 bonus action for this casting.
Quicken Spell says it is changing the casting time. War Caster isn't saying that. It instead is bypassing the casting time, allowing you to use your reaction to cast it. The Sage Advice Compendium, under Can you use a melee spell attack to make an opportunity attack explains this:
Each spell has a casting time. (...) The War Caster
feat is an example of a feature that does let you bypass a 1-action casting time to cast a spell in place of making an opportunity attack.
This means more generally that the casting time of the spell (which is a property of the spell), and the action required for casting it can be two different things.