I may be wrong but historically the rule of thumb used to be a person doing physical door knocking, asking face to face for work, was considered to be more likely to pull their weight and get stuck into a job, compared to an anonymous letter writer, expecting a prospective employer to contact them back. (obviously some of this goes back to the days of factories which typically had signs on the gate stating what vacancies were available). But to try and put this into context of modern communication, sending random incorrectly addressed e-mails with attached 20 page PDF CV is most likely going to go straight into the delete & trash pile. Too many people try to waffle and blag their way with quantity over quality, but if you are an employer and you get 250+ email job applications with pages and pages of irrelevant waffle about wiring up your grandma's pantry light, your uncle Bobs loft socket and the next door neighbours garden pond pump & light, in reality no one is going to read much past the first few paragraphs before deciding if it is bin material.
"First Impressions" was and still is a fundamental governing factor to if further interest, communication or activity will occur. You have got to think about what are the key selling points to make a prospective employer interested in you. Advertising is and was pretty much the same, (and hasn't changed), for Cars, TV's, Phones, Clothes, Holidays, Make-up, Ready-Meals, Fitted wardrobes, Garden tools, Kitchen appliances, etc, etc, And persons wanting to sell themselves to be the preferred choice over the competition. For any job vacancy you must use similar techniques to those used by successful advertising. Get the fundamental key selling points covered on the opening front page with pointers and direction to any relevant more comprehensive detail that can be read further on later.
Some companies have websites that include pages about their key staff, including names and their job role within the company. If you can google a prospective employer, then address any letters, emails, phone calls, or physical visits, asking to speak to a specific persons e.g. Mr/Mrs/Miss Clarke (Head of recruitment / Engineering manager etc), it is more likely to at least be read and/or evaluated than a bland 'Dear Sirs' email sent to contact form at Widget&Sons.co.uk.
It sounds to me as though you are just going to have to start a long hard graft, pushing and trying any and every door, searching for anything from apprentice to mate to tea boy to cleaner! If anyone says they have no vacancies at the moment, thank them very much for considering you, then ask again a few months later in-case circumstances have changed. Do NOT give up trying as the construction industry and all of its associated services can have a high level of flux and change.
Doc H.