Since you're trying to do two things in one turn that both require an action (casting Blade Ward and making a Dexterity(Sleight of hand) check), there are basically a few approaches. You'll either need two actions in a turn, or convert one of those two into a bonus action.
Here's some things you can do:
Taking two actions in a turn
There's only one thing that gives you two genuine actions per turn, and that's Action Surge. It requires two levels of Fighter and access to the Blade Ward cantrip. You can get Blade Ward from the Magic Initiate feat, so a variant human could perform this trick once per short rest at level 2, which is also the lowest level it can be done. (But you'll have to be a Human Fighter with good dex and int).
At level 3, you could become an Eldritch Knight and pick up Blade Ward from there; you'll probably still want to be a Dex/Int Fighter, but it's not a hard requirement.
The only other thing I know that gives you an extra Action is the Haste spell, but that action is restricted and allows neither an ability check nor a Cast a Spell, so it's not going to help.
Making the Sleight of Hand a bonus action
This can be done by the Thief, as you pointed out in your question. That means you need 3 levels of Rogue and the Thief archetype, and access to the Blade Ward cantrip. Normally if a Rogue wants a Wizard Cantrip, it's easiest to become an Arcane Trickster, but that option is off the table. In this case, you'll need Magic Initiate to do it, which means either Variant Human or being level 4 and taking the Feat.
This option is probably the best one, because you can cast Blade Ward and then pickpocket all day long.
Making Blade Ward a bonus action
You can be a Sorcerer and pick up Quicken Spell, which allows you to cast Blade Ward as a bonus action a few times per day. Blade Ward is a Sorcerer spell, so you could perform your combo starting from level 3, when you can take your first Metamagic.
At that level, you'll probably only be able to do it once per day, though. It grows over time, and you can sacrifice more spells to do it more often, but it's going to be pretty limited and expensive in terms of Sorcery points.
It has the advantage that Quicken Spell is a cool either way and you'll want Dex anyway, so this path locks down almost nothing about your other character options.