The Evening Standard ran the headling Move over Amsterdam, the London cycling revolution is in top gear. Quotes:
‘The study for City Hall reveals that Theobalds Road, Holborn, is London’s busiest bike street as 64 per cent of vehicles passing along it in the morning peak are bikes, followed by Kennington Park Road, which runs between Kennington and Oval (57 per cent) and Old Street, Shoreditch (49 per cent).’
‘Blackfriars, Waterloo and London bridges are all now among the top 10 busiest cycle streets in London. On all of these, cyclists make up 42 per cent of traffic and 15 per cent of people – though they take up just 12 per cent of road space.’
‘Almost 9,300 riders – 11 a minute – cross London Bridge a day. Along Amsterdam’s busiest cycle route through the Rijksmuseum there is a daily frequency of 13,000.’
But the Netherlands spend £30/person/year on cycling and the UK spends £2.20/person/year on cycling. That makes a difference (figures from 2013-08-07 BBC Newsnight Cycling clip).

The problem is that Openstreetmap requires more coding skills than non-specialists are likely to have. School geography departments could do a lot of good by teaching GIS instead of the classic ‘exports of Brazil’ curriculum. In the UK, IT teaching in schools is about 30 years behind where it should be – largely because school teachers, even in IT departments, lack coding skills. They teach IT as though they were teaching anatomy ‘this is a VDU, this is a CPU, this is an HDD’ etc. HOWEVER, a start has been made on using OSM data to produce a map of cycle route use intetnsity in London.
I agree with London’s first cycling commissioner (Andrew Gilligan) that ‘The point about almost all “cycle infrastructure” in London is that it is not designed for cyclists. It is designed so that politicians can say that something is being done for us. Perhaps 10 per cent of cycling schemes are worth having. The rest range from pointless, to ludicrously bad, to actively dangerous. At various points along its length, Cycle Superhighway 7 hits all four categories.’

I think of former ‘horse and cart’ routes as being rural but you are right that they also exist in towns (sometimes a former ‘horse and carriage’ routes ie boulevards etc). Where they have not been widened or re-aligned they also make good cycle routes in towns – but most of them, like Oxford Street, Fleet Street etc in London, have changed out of all recognition. However it would be possible for a surveyor, making use of old maps, to find urban roads which are still reasonably close to their form in the days of horse-drawn vehicles and my guess is that they would be good cycle routes. Blackheath High Street is a local example and I shudder to recall that in the heyday of ‘modernising London’ they hoped to make it into a four lane road. I would now like to see it returned to its historic condition as a ‘shared street’ by making it more friendly to pedestrians and cyclists and less friendly to motor vehicles.
Your history project would be excellent and my suggestion is that it should be done using Openstreetmap data AND with similar funding – by which I mean no funding: the work would be done by volunteers. Openstreetmap already has better information on footpaths and cycleroutes than government mapping agencies.

You would probably need to do three different surveys: one of existing roads, one of existing cycle paths and one of where cyclists really want to go (their ideal map).

So it would be good to have maps for the first purpose of understanding existing roads showing:

1) the period of history which all the roads were from, ie Roman roads, Viking roads, medieval roads, renaissance roads etc.

2) A description of the roads from each period of history ie. typical width, length, divisons by features and other typical characteristics, both in plan and section.

3) A map showing relevant topography, ie flat, uphill, downhill, across a slope etc.

There is a range of London cycling maps but, like all maps, they are made for a purpose and that purpose is NOT for helping cyclists get from A to B. Most of them are propaganda and what they say is ‘we care about cycling’. I try the routes from time to time and find them (1) very difficult to follow (2) slower than motor roads. It would be good to have a survey of cycle traffic on all roads so that (1) comparisons could be made between these stupid routes and what cyclists actually do (2) expenditure on improving cycle routes could be targeted on where there is most demand.

The word ‘road’ is etymologically ‘a place to ride’ and riding on a horse was the main transport mode, for the rich, from Celtic times (maybe 3000BC) until c1700. There were carts for transporting goods but not many roads which could accommodate them. Then there was a great expansion of wheeled traffic. The wheels were bicycle-size and bigger and the geometry of these tracks was perfect for cyclists: excellent horizontal curves and vertical curves. So these are the tracks which should become cycle paths – and none of them should ever be modernised!

Is there a difference in trying to accommodate cycling in Roman or Medieval London as opposed to the newer areas?

It would be interesting to consider the history of transport and how it was accommodated. Presumably walking was the commonest mode of transport in Roman times. I am not sure when horses and carriages became the most common form of transport? But I am assuming that cars date mostly from the interwar period of the twentieth century?

Each of these changes would have transformed the nature of public thoroughfares.