I should state at the outset that IANAL.
This is what CEC says: https://www.edinburgh.gov.uk/roads-travel-parking/street-schemes-proposals/4
This is the The Local Authorities' Traffic Orders (Procedure) (Scotland) Regulations 1999 referenced on that web page: https://www.legislation.gov.uk/uksi/1999/614/contents/made. Section 20 is the bit that applies specifically to experimental traffic orders: it says that sections 5, 6 & 7 of the regulations (publication, notice & objections) do not apply to experimental traffic orders.
This is the Road Traffic Regulation Act 1984 which underpins the above regulations. It's the one referred to in the above regulations as "the Act". Section 9 is specifically relevant to experimental traffic orders. Interestingly, Section 9(4) says that an experimental traffic order can be extended for a further 18 months (subject to certain conditions which I don't have time to check right now).
Have fun. (And if you think trying to untangle this stuff is complicated, you should try the regulations surrounding traffic signs.)