(Posting this after checking it in patch 3)
If you multiclass as Cleric into a martial class and gain the Extra Attack feature your character will automatically use up your bonus action and War Priest charge before using the regular Extra Attack. This does not happen if you gain the War Priest feature
AFTER you gain the Extra Attack feature. If you gained War Priest first then every attack action will automatically default to using the bonus action first before using the martial Extra Attack.
You can circumvent this by spending the bonus action before attacking for a second time
EVERY SINGLE TIME YOU DO NOT WANT TO SPEND YOUR BONUS ACTION AND WAR PRIEST CHARGE but it can be easy to forget that.
Alternatively, you can respecc and level the class that gives you Extra Attack first and then multiclassing into Cleric.
This has a) the downside of proficiencies being different and requiring a respecc and b) This still doesn't work with Haste/Action Surge. Your Character will default to using the War Priest extra attack rather than the second Action granted by Haste/Action Surge (both if you do have a martial Extra Attack and if you do not).
Additionally, in the (very unlikely) scenario that you leveled to 5 as Fighter picked 1 level of War Cleric and then went to Fighter 11 and gain Improved Extra Attack your character will attack first with the action, then with the War Cleric feature and then with the 2 attacks from Improved Extra Attack.
It would be nice if the War Cleric Extra Attack was a passive feature you can toggle on/off like Repelling Blast/Non-Lethal-Attacks or better yet like Quickspell Gloves.
https://bg3.wiki/wiki/Quickspell_Gloves