Coroners Officers will be your first point of call. It's up to you to choose whatever funeral director you want so don't feel pressurised into choosing someone - ring around when you are feeling up to it to get some ideas about what they will charge you and go with whomever you feel comfortable with. There is nothing stopping you choosing someone from your area. You may need an Out Of England certificate (not sure on the rules when it's Scotland), but the Coroners Officers will normally arrange that for you and it will be sent to the appropriate mortuary.
Usually the body will be released from the Coroners Officers point of view after the post mortem and the subsequent findings. If nothing is found then it can take a couple of extra days for tests to be run, but try not to worry, this often happens so don't let it get to you or anything. The Coroners Officers will arrange for the death certificate to be issued (around here it's called an B form but may be different in Scotland), this is sent to the registrars for you where you will need to go to officially record the death.
From your funeral directors point of view, they will probably go to collect the body and drive it back, that is how most of the funeral directors work around here. They may subcontract someone else to do it, but they will also be a funeral director, and that is quite common to happen this way as well.
If you want any more advice away from the forum just PM me, I'm a anatomical pathology technician in a mortuary so know how most of this stuff works. Just take your time over things and don't rush into anything, it will eventually become easier for you.