I have received an impressively comprehensive response to this thread (via Andrew Burns).
"
have collated the information provided by the New Works Team, City Centre and South Neighbourhoods below.
Buccleuch Terrace and Bruntsfield Place – New Works Team
The ponding issue highlighted as “…end of The Meadows… new route to the Innocent” is the same issue as that of the last location noted in the attached PDF. I can confirm that work is nearing completion to install an additional gully at this location, which should remove the water build up following the installation of the new cycle route.
On Bruntsfield Place, and following significant resurfacing works, the contractor has been asked to investigate the sections that are currently ponding and to provide a solution. You can contact New Works Customer Care via transport.newworks@edinburgh.gov.uk for further updates.
North Bridge
As this location has a significant traffic volume, including numerous bus services it is difficult to provide a solution without causing major disruption to the city centre. The City Centre Roads Team are currently seeking a permanent solution, with a view to completing works prior to the start of The Edinburgh Festival. However, it should be noted that remedial works would require special traffic management and could conflict with other priority works. In the mean time a temporary repair will be progressed at the low point to try and lessen any water build up.
West Nicolson Street
This ponding was attributed to a blocked road gully. The gully was last attended on 1 April 2015 and reported as working, however and following inspection we have notified the Roads Gully Team, requesting further investigation and action as appropriate.
Leith Walk
The ponding issue adjacent to the Casselbank Street junction on Leith Walk has been highlighted with the contractor responsible, with a view to rectify water build up. The contractor has surveyed their works and will be submitting remedial proposals to the Council. Further information can be obtained from the City Centre Roads Team via roads.centralvip@edinburgh.gov.uk.
Conference Square
The area is owned by this Council but is not adopted as part of the roads network. Management of the space is undertaken by Estates Services.
The Council is aware of the ponding issue and has been working closely with Standard Life and Sheraton Hotel, who both have an interest in this location.
The problem arises from a defective and somewhat complex drainage system beneath the Square. A jointly commissioned survey and some clearance works have been undertaken. From this survey, consultants have advised of a solution which will involve a significant re-design of the drainage network.
The Council is now examining the arrangements required to proceed, and is engaging further with affected proprietors to progress the necessary works.
"