Maths and Physics teachers are one aspect of encouraging young people into Engineering, although typically these teachers do not know about the whole range of career opportunities in Egineering and think, as the media does, that only Dyson and product design is engineering or, that it is the personwho repairs a washing machine . Most maths and physics teachers do not understand chemical engineering, environmental engineering, civil engineering, mechanical engineering, computer engineering and so on. Careers advice for engineering subjects is woeful. My DH was lucky enough, many years ago, to do an Engineering A level. He sailed through Y1 of university. Engineering in schools (Technology?) has been dumbed down so much that young people never get a real taste of what they could be doing. My DH also did an Architecture course whilst at school and has lessons at a university in their Engineering Department. No-one is remotely inspirational in schools regarding how to motivate young people to be Engineers. The curriculum is only geared towards academic maths and physics, or some sort of technical curriculum that does not prepare for rigorous Engineering courses at university.
Also, graduates from the many of the top universities do not go into engineering jobs. They go into City jobs, in finance and managment consultancy. This exacerbates the shortage. Teaching, and not even fairly well paid graduate engineering jobs, cannot hope to steer them away from far more lucrative careers in the city. The truth is that lots of young people do not want to be Engineers because they do not know what the opportunities are and they are not stimulated at school into thinking about this career. No amount of maths teachers and physics teachers will change this unless society changes its views on the status of Engineers and engineering opportunities are the norm in schools.
I also think you really have to want to be an engineer and have the necessary skills and attributes, not just STEM subjects, but problem solving, taking things to peices, learning how to construct machines and buildings, understanding the issues that Engineers need to solve, etc. You really cannot persuade arts people to swap into engineering if their heart is not in it and it makes no difference as to who is teaching them!
Also lots of schools, and the public, do not understand that Chartered Engineers and Engineering graduates are not the people who fix your washing machine! We do not give respect to Chartered Engineers in this country and the pay for many is not fantastic after 5 or 10 years. It takes many more years to be a Chartered Engineer than it does a teacher, by the way. Both suffer from being perceived as less than glamorous and with capped earnings that do not match qualification and effort.
Rather long message and probably off topic - sorry!