CP this is a most worthy cause!
I thought perhaps I would describe, from what I gleaned when I was in industry, how lobbying for change within government was done. Since the context of what I used to be involved in is different, I must stress I am not saying this is how we should do it now, but just in case some experience there is useful for reference.
A campaign would likely be in two phases.
The first phase is to get as many ideas as possible on the table.
But to increase the chance of success, the second phase needs to be structured, and it helps to start with a vision of the end in mind. For the sake of discussion, the end is perhaps a commitment by a key decision maker, who is likely to be a politician who may or may not be the PM, that leads to effective action, which might be legislation.
The thought process and work might then be something like this:
1) accepting that politicians will be more likely to support what is good for their popularity/electoral success, develop a straw man (a draft proposal that is to be shot at and improved on) that firstly illuminates alignment to that interest as well as for the greater good, and secondly addresses how opposition from interest groups who will be negatively impacted could be countered effectively by the decision maker. Simply saying what we want and what is good is natural, but might undersell our viewpoint.
2) identify potential supportive influencers and experts, perhaps ministers in the Department of Transport, the Mayor or well placed professional lobbyists who are regular cyclists, and find out if they are willing to give us some time individually to discuss and advice. The success or failure of something like this is partly the proposal, partly how it is delivered, and partly what other supporting voices can be corralled to bear on the decision maker. Just a letter to the PM might work, but might be more dependent on luck than is necessary.
3) Based on what is known and advice received, work like mad to deliver the right message and the right environment for reception at the right time.
As I said I am not saying the above is the process we should adopt here, but was ime how campaigns were mounted by a multi billion £ corporation. For example, any politician who wishes to legislate for cyclists might worry about alienating a greater population of drivers, it might help if we could propose a solution that navigates that. I am no expert, but depending on what the law says perhaps only a relatively small change in policy in the department of the Director of Public Prosecution which is under the Attorney General would do the trick, hence a clear vision of the end at the beginning and expert advice along the way could be important.
Just my 2 cents. Would be delighted to help further whichever way I can.