If you are extending out the back of anything that is an extension you'll need planning permission. It looks like the right 1/3rd of the house may be an extension so as you plan to extend out the back of it you'll need PP.
We are going through exactly the same process now. Below is roughly what you need to do:
1) Speak with an architect to draw you detailed plans of the exterior of the extension, work with them until you are happy with everything
2) <Optional> Go and sweet talk your neighbours, explain you'll be getting an extension here is the rough plan and details. Once you apply for planning they will all get letters from the planning office with the details anyway so if you get them onside before hand it makes it much easier.
3) Submit drawings to planning office via your architect. PO may come back with suggestions, demand alterations and take feedback from neighbours. This part of the process can take up to 3 months (and it likely will be close to that)
4) Once planning is approved you can legally build it, you have 3 years to get started (break ground) before the planning needs to be applied for again. As you can now legally build your plans you'll want to speak to a Structural Engineer who will be able to calculate the materials you need (such as RSJ's as it looks like you'll be removing load-bearing walls). Your architect will likely be able to recommend a SE to carry this out. You'll need to work with the SE as it will cost more to hide RSJs and they may need to make suggestions about beams and various bits. The SE will ensure everything is going to pass building control (who ensure the building is safe). Building control will often make a few visits during the build to ensure everything is being done as instructed by the SE. Building control do not concern themselves with anything to do with planning (which is about if you can legally build it), they are just making sure the right materials are being used and everything is safe.
5) Once you have the drawings (architect) and calculations (Structural Engineer) you can give these to a builder for a formal written quote.
6) Once you have picked a builder you can get started.
It's important to remember that planning and building control are two completely separate things, neither cares about the other. Architect and planning office is purely how the building will look from the outside and if you can legally build it. Structural engineer and building control are there to make sure everything stays up, is safe and properly supported.