In terms of ratifying the WHC, my understanding is that Scotland would have to do this separate to the UK. I can't imagine that causing any particular issues and that is what the 18 months between the referendum and independence is for.
An independent Scotland could not of course ratify the Convention until it was officially independent (and was either invited to do so or had joined UNESCO).
I had a quick look at the example of the Czech and Slovak republics
a. Slovak Declaration of Independence -17 July 1992
b. Agreement for a peaceful dissolution of Czechoslovakia - 13 Nov, agreement on division of property and 25 Nov 1992 - New Constitution Act setting date etc. (A lot faster than the Scottish break which would take 18 months!!)
c. "Velvet Divorce" - 1 Jan 1993. We know from this and other examples (Eg USSR) that new countries "carry forward" inscribed sites even if they haven't ratified the WH Convention
d. Joined UN - Both on 19 Jan 1993
e. Joined UNESCO - Slovakia 9 Feb, Czech Republic 22 Feb 1993
f. Ratified the World Heritage Convention - Czech Republic 26 Mar, Slovakia 31 Mar 1993
g. First New T List - Czech Republic 19 Jan 2001, Slovakia 12 June 2002 (as both made further nominations and gained further inscriptions during this period it implies that each "carried forward" its part of the Czechoslovak T List
h. First New Inscription Czech Republic - St John of Nepenuk 1994, Slovakia - Banska Stiavnika, Levoca and Vikolinec all in 1993 (which means their nomination must have been started before the Velvet Divorce of course)