I tried my best to design your apartment according to your instructions and desires, and I hope I managed to do so.
First of all, I took the liberty to design your space a little bit different from most of your reference photos. I took a more contemporary approach, opposed to the classic one on your references, but Scandinavian style remains!
You expressed a wish to somehow divide the hallway from the living area. I proposed a custom-made room divider, and in this way, I managed to avoid adding a wall. The room divider separates the space, while it also serves as storage, cloth hanger, shoe rack and shelving unit.
The only wall that I proposed adding, is the one between kitchen and guest room. I tried to balance the dimensions of both this spaces, so that neither of them is too big or too small, and they contain all of the furniture pieces and appliances that you have requested.
Since the day-light is a major issue here, my design was driven by a need to brighten up the space as much as possible. For this reason, I used a lot of white color for its virtue to reflect light, and some wood texture to bring some warmth into the space. As you can see on the renders, I also proposed changing the existing patio door with a sliding door, for the same reason. But, you should consult a civil engineer prior to doing this, because it is possible that the door opening contains some load bearing elements.
Most of the furniture pieces are industrial products. For the ones that are supposed to be custom-made, I provided detailed drawings, except for toilette shelving unit, for which exact measures were not provided. This shelf was proposed to hide the unappealing space behind washing machine. It is not complex, and any carpenter can figure it out from the information I provided.
And finally, about the budget. Sorry, but I couldn't find any relevant prices for custom-made furniture in Great Britain. The rest of the costs are estimated to 18.000 GBPs. Adding this custom-made pieces to that sum, your budget of 20.000 will probably be slightly exceeded. However, the new budget includes upper bathroom, which was not included in your initial budget. So, it fits!