The steps to changing the bike tube without tire levers: In the first step, you should have a deflated tire because it is much easier to squeeze its sides. The second step is to loosen the beads. Get one tire bead into the center of the rim by pushing the tire into the bike’s rim. These beads will act as the primary support to ultimately push the tire off the bike rim. Start using your thumbs to push the tire bead off the bike rim. Then, push the rest of the bike tire away. The next step is to inflate the tube. When your tube has some air inside, which will definitely help retain its roundedness and proper shape. After this step, you might Pop the Valve and Insert the New Internal Tube: lay the tube over one side of your tire and align the valve bike stem with the stem hole. Then, carefully squeeze the tire where the hole is located and put the valve stem through the valve stem hole, pull on the bike stem to ensure that most parts of it go through the hole as much as possible. Finally, push the bike tire wall back into the rim.
This guy in video above makes it look easy. I find that the tube needs to be fully deflated ( and sometimes valve removed) for it to work. I tend to use levers to remove tyre as I tend to leave the wheel on the bike. I never use leavers to put back on the tyre.