Nothing will happen to either relationship, the only consequence that comes along with dating while married is that your spouse will have a chance to become jealous/upset if you gift someone you're dating.
There's also the group 10 heart events (that trigger if you're dating all of the bachelors, or all of the bachelorettes) which causes the respective set of bachelors/bachelorettes to be upset with you for a week, but that doesn't come with any consequences to friendship level.
If you are modding already and want to marry both Alex and Elliot, I think there are polyamory mods? But I've never used nor looked into it.