Hi @tombot1974,
Thank you for your question. This looks like a very interesting mini-project that I'm excited to see you bring to life.
Connecting the two vertical panels with the bottom two flat panels will be fairly simple. Use 2 screws on each side of the horizontal panels that are long enough to drive through the vertical panels and embed about 50mm into the horizontal panels. Depending on the width of your vertical panels, these Zenith 10 - 8 x 100mm Galvanised Countersunk Rib Head Timber Screws should be a good choice. If your vertical panels are wider, use a longer screw.
If you'd like to add more strength, you could use timber dowels between your vertical and flat horizontal panels. They can be glued in place with wood glue.
Check out How to build simple woodworking joints for some info on dowel joints.
For the shelf, stopping the shelf from rotating when weight is applied is the main issue we have to deal with. This is tricky because to stop rotation with fixings alone, we need a minimum of 2. The thickness of the timber in your inspiration photo does not look to be enough to achieve this.
To prevent this rotation, I'd suggest cutting 20mm notches on each vertical panel for the shelf to sit in. The shelf can then be fixed in place with the same long screws I have mentioned above.
You can also use thin, flat brackets such as these Carinya 20 x 200 x 2mm Flat Make-A-Bracket on the back of the shelf to connect to the vertical pieces. You can use small Button Head Stitching Screws to keep a low profile.
Let me know what you think and if you have further questions, please don't hesitate to ask.
Jacob
