First of all unless you have other facilities elsewhere in your property you may have to do without a toilet etc. for a couple of days until the job is done. As for the work entailed, I presume the outlet for your existing toilet goes outside the property which means you have to break out or cut into this existing toilet outlet pipe, probably outside your property, in order to connect and extend a new soil pipe up the outside wall to the room above. If this can be done it is likely to be the most difficult part of the job and should not be tackled by an amateur.
A competent plumber should be able to complete the works in 2 to 3 days. Difficult to give estimates without seeing job but get at least three quotes and don't necessarily accept the cheapest one. If you can get some background info on the plumber you select you should be OK. There are some good ones about, just be careful.
Re your conservatory, you say there is a drain outside. If this is just a gully which your kitchen sink waste pipe discharges into it should be fairly easy to seal and box in, just make sure you can get access to this drain if necessary. Whoever does your conservatory will almost certainly have come across this problem many times.
If you type the key words of your question into our search box to the left of the site you may find the answer is already posted or is in the DIY projects section of the website. Every post goes through a monitoring process and using the search box may speed up your answer.
DIY how to tutorial projects and guides - Did you know we have a DIY Projects section? Well, if no, then we certainly do! Within this area of our site have literally hundreds of how-to guides and tutorials that cover a huge range of home improvement tasks. Each page also comes with pictures and a video to make completing those jobs even easier!