Prepare for your escape when the jacket is being put on. Expand your chest as much as you can and hold elbows tightly to your sides when the jacket is tied around you. If possible, cross your dominant hand over the other (right over left if you're right handed). The right hand should ideally end up on the left bicep or ender the left elbow. The more tightly the arms are folded at the chest, the more difficult escape will be. Fabric can be also gathered in both hands under the armholes where it won't be noticed to buy a little more wiggle room.
Escape by breathing all of the extra air required to expand your chest and relaxing all the muscles that were tightened and held rigidly. Scrunch the shoulders forward and shrug shoulders as deeply as possible into the sleeves, loosening the back of the jacket.
Use this slack to work the arm's buckle up to the point where they can be lifted over the head or down to where arms can be stepped over. Standing with the elbow up against a wall or another stationary object will help you move the buckle as will getting down on one knee and using the other as a brace.
Use the hands, which now have space to move a bit, to work at the arm ties (or straps). Sometimes if there is ample room, the ties or buckles can be undone with the teeth. Once the hands have been freed, they will be in a position to undo any further straps.
Practice until you get it down, but remember that strait jackets are not designed to be worn for extended periods of time. They become excruciatingly painful as blood pools and swells up the elbows inside the tight sleeves. Shoulder, back and neck muscle cramp painfully and can't be moved to ease the pain.