Almost certainly balanced, possibly underpowered
It seems that the intent is to give a melee caster some damage that can affect creatures resistant or immune to non-magical weapons (or the fire and thunder types of the other "blade" cantrips). In that case, you could simply make the effect "the weapon is considered magical". Force damage, while being the least resisted, is still more resisted than straight up magical weapon damage, I believe.
So, we have a couple points of comparison. The second level Magic Weapon, which also gives the weapon a +1 to hit and damage, and lasts much longer, and can be used on an ally's weapon.
The Druid cantrip Shillelagh, which has a longer duration, potentially increases the damage to a d8, but still requires using the spellcasting ability modifier to attack and is limited to clubs and staves. It also takes a bonus action to cast, which lowers the action economy for the first round (taking a bonus action and an action to attack), but then maintains better action economy subsequently, since it allows multiple attacks per round via extra attack and two-weapon fighting. It does not scale the damage as the caster levels, making it less useful (and possibly useless) after the caster gets a magic weapon (a d6 damage weapon with +1 to damage being effectively equivalent to a d8 weapon with no bonus as Shillelagh provides).
So, for the most part, it is superior to Shillelagh, inferior to Magic Weapon (as it should be, though since it would stack, you could use both), and quite possibly inferior to Green-Flame and Booming Blade in most situations, since at high levels the user should have a magic weapon and thus not suffer from an opponent's resistances, and at low levels very few opponents would have those resistances. There are, of course other possible comparisons, but I think these are sufficient to determine the spell's appropriateness.
In short, I feel that it is niche enough that it is probably under powered, but since its damage scales, it could certainly continue to see use even after the caster has a magic weapon, especially since the bonus damage type will be resisted much less than that of GFB and BB.
In fact, that might make the best comparison of all - you lose some conditional damage and use a smaller die for the bonus damage in order to get a better damage type. One or the other of those might be sufficient to make it balanced, but with both it is certainly in little danger of being overpowered.