How is infrastructure for bicyclists designed, and how does it evolve? Question: How is infrastructure for bicyclists designed, and how does it evolve? 

Related questions to prompt more detailed answers:


When is the right time to build bicycle roads?
How is the budget evaluated for new bicycle facilities? 
Is there some way to verify the budgeting premises for bicycle projects?
How are different facilities evaluated? 
Are there stages which a city goes through to get a superb bicycle infrastructure?


Longer term considerations:


Suppose the fundamental city plan for cargos and cars is to produce a smooth flow of traffic. Is it easier to plan long-term bicycle road investments together with other roads?
You cannot take every point into account at the start, so how do we make improvements to existing facilities?
How do we make sure progress doesn't stop or reverse?
Is there a way to avoid only reactive fixes (something gets improved only after fatal accident)? 
Is the improvement chaotic or a linear progression?
In most places the answer to your main question is "very bureaucratically", usually involving multiple government departments and a lot of competing interests. It's a part of town planning, which is itself contentious in most places. Even without the sort of natural input that, for example, Christchurch is currently experiencing.

My experience is that even when national governments are involved detailed planning tends to be done at a very local level. This contrasts with major roads where there is usually a government department who is responsible for them. So national funding ends up in the hands of local government who design, build and maintain bicycle facilities in their area.

Ideally the work is overseen by someone with both qualifications and experience in designing good cycle facilities, with public involvement during the scoping and design stages. This is when cycling organisations can be very influential both in terms of what route is chosen and the detailed design that can make it a stunning success or a miserable failure.

Budget works two ways. The "how much do we have to spend" is usually a political question and is influenced primarily by lobbying your government representatives. The "how much does it cost" is in theory an engineering question, but is often determined by political considerations, specifically the "it must be able to support heavy maintenance vehicles" requirement that can mean a cycle facility costs the same as a standard roadway instead of less than 1/4 as much. Local government and roading engineers are usually very keen on this for two reasons: it gives them a "free" access route (they don't pay for it, cyclists do), and it pushes up the apparent cost of cycle facilities (so they have to deal with fewer of them per dollar in the budget). In some situations this does work the other way, where the builders go "cycleways are cheap, let's put them everywhere". Smart property developers do this as they can still call them roads (and they look like roads on the plans), but they're cheap to build.

Evaluation is complex because there are many different ways a cycle facility can be evaluated. From how good it is to ride on, to accident rates or population health, to maintenance costs, to political benefit (or cost), just about everyone will have their own way to evaluate a cycle facility. This is a contentious issue that my local BUG addresses with our local council almost every year. Even though we have a cycling team in the council infrastructure department we have to fight to keep them as well as maintain existing facilities and build new ones.

The broad progression that I'm used to is: 


no cycle facilities. 
national government makes funds available to local councils for cycle facilities, and local government wants those funds. 
they paint "cycle facilities" onto existing roads and build a few recreational facilities to get that money. 
local cyclists are pressing for better facilities. 
Eventually council appoint a part-time "cyclist appeasement officer" to deal with them, who also has a budget. This gets spent on making cyclists more visible, usually by printing maps of the facilities and flyers saying how good cycling is. 
Over time the budget increases and the focus changes to off-road facilities that actually go places. 
a council employee in the road-making department is designated as something to do with bicycles ("cycle facilities coordinator" or similar), 
eventually this office gets onto the list of "people who have to be consulted before road changes happen". 
Around this time cycling becomes something that politicians mention during elections
over time politicians settle more onto the pro-cycling side. 
a mayor/governor who is a cyclist is elected and suddenly a key question for every single transport project becomes "how does this help cyclists?" 
Over time this becomes the dominant paradigm across the nation, 
eventually Dutch is legislated as one of the national languages.


(note that places move both up and down the list over time - it's not all win-win-win)

A few local examples. This is my councils transport page featuring bicycles and pedestrians with their own headings. If you read the strategy you can see that the local BUG had some effect. Also note the "sustainable transport" menu of the left of that page. In Sydney where I used to live they now have a "Manager, Cycling Strategy" and a heap of content on their web site.