How to contact casting agents for large UK theatres
When considering your approach to a theatre you have no prior relationship with, their website is the best place to start. A lot of theatres who produce their own work will have instructions to actors telling them how to approach the company in the 'Contact' or 'About Us' section, so look there in the first instance. Always follow these instructions if they are there, or you risk annoying the company or making it look like you haven't done your research. This isn't a good first impression so it's advisable to do things the way they like them done, they will have a reason for it.
The theatres will generally fall into two categories; those who handle the casting internally and those who employ external freelance casting directors. If this information isn't on the website, it's ok to call and ask how they prefer to be approached by actors, just keep the conversation brief, friendly and professional - plan what you are going to say before you pick up the phone to avoid stuttering and awkward pauses. It can seem a bit daunting at first I know, but as time goes on you will begin to find that you know how individual theatres/casting directors like to be approached from past experience, you just need to get to know them and how the operate.
If the person who handles the casting is full time with the company you can just write to them at the company address (or e-mail if that's what they prefer), if it is an external casting director then have a look through Contacts or The Actors Yearbook, you can find both available to order online, or give them a google to find their contact details. Again, you will need to check how they like to be contacted.
After that it's all about thoroughly researching the play that they are casting and making a sensible suggestion. Only suggest yourself if you know you are right for the part. General applications along the lines of "If anything comes up that you think I might be suitable for please get in touch" are not welcomed. If you have any specialist skills that you think would be useful for the production then make sure you highlight them in your letter (A guy in Romeo and Juliet for example will probably have to do a bit of stage combat). It's also worth mentioning any link you may have to the theatre. If you are from the area or have family there, or have seen a lot of the company's work and can remember something you particularly enjoyed it can be worth adding in. If you haven't seen any of their work that would be a good thing to remedy. Go and see something, get a feel for the kind of work that they do or the style they use and figure out if and where you would fit in.
Obviously your application should include a C.V, a 10 x 8 professional headshot and a well thought out, well laid out and specific covering letter. This is standard. You may also have a showreel, but for theatre productions these are not useful and should not be included in your application.
I think calling to confirm the application has been received is a waste of time and a bad idea as it could irritate, but some people do it. If you choose to do it give it a week or two first and then call, but don't pester.
If they want to see you they will call, if not don't worry, just move on to the next application, you will be called in eventually.