I've been in a similar position. I work for a multi-national company and there have been a few responsibilities that I've had to complete for a company that was located in UK. (I work in the Eastern, US.) The team in the UK would often send me emails requesting that I URGENTLY start and complete some project for them. I'd quickly respond with an email, requesting necessary follow-up information. After a couple of days with no response, I'd send another email. Yet again, no response. Below is the strategies that I've had to employ while working with this company. You may not complete your projects, but following these guidelines will help you both keep your sanity, while track your efforts to complete your work.
As you've stated, try to keep your initial email brief. We are all busy, so writing a thesis of any trouble or issue could cause avoidance of the problem. If something needs to be thoroughly discussed, it might be worth sending an initial email, requesting a phone call to discuss the issue(s) further. Then you can discuss at length the specific concerns that you may have or get the finer details that you need to advance your project.
Wait a couple days, to a week, depending on the level of urgency. Remember, you are working for them. If they don't respond, it might not be as urgent as they might have initially stated. When you respond, open your original email and respond to it. Do not write a second email from a blank, new email. Of course, you need to change the recipient from yourself to the project manager. (I've forgotten this on a few occassions and I've sent myself emails intended for others. o_O)
The reason you respond to your own email is because this creates an e-paper trail that you are trying to further the discussion to the best of your ability. This is important, as you will see soon.
After sending the second email, wait again for another few days to a week.
After not receiving a response, try to contact the primary contact over the phone. If they are unavailable and you get a voice mail, hang up. Yes, that's correct. Voice mails are easily lost and are untracable. I, therefore, avoid leaving messages over the phone unless I am absolutely positive that I will get a response from this specific person.
Instead, start calling other individuals who are next in line of responsibility for this project. If you can't get ahold of them, think of anyone who might be loosely related to the project. See if you can speak with them directly and have a casual discussion of the situation.
They may not be able to help you finish your work but they might be aware of any circumstances that have changed. They may be able to tell you your manager is away on a vacation, or if other circumstances have kept them away from everything associated with your project. The bottom line, try to find some one who can make any kind of statement about the status of this project, even if it's not their responsibility.
Once again, respond as a reply via email to the project manager and state any information relayed to you from the person you managed to speak with over the phone. Tell them that you are tabling the project until he or she directly contacts you. If the individual that tried to update you on the status of your project was incorrect, the manager should be frustrated enough to contact you and clear up any confusion. This is a bit of social engineering, but this technique has actually opened up channels of communication for me in the past. It's at least worth a shot.
If you can't manage to get anyone on the phone who is useful and the company has a receptionist, speak with them and see if the project manager is on vacation or away on business. It's been my experience that compared to us Americans, the Britains take quite a few "Holidays" and are some times gone for 2-3 weeks. This could be true in your circumstances as well and during this time period, you shouldn't expect any form of response.
If this is the case, just defer any further attempts at communication until the time in which your PM is suppose to return. Give yourself one more shot at communicating with them. Keep in mind, if they've been gone for a few weeks, they will probably have a ton of emails to sort through when they get back in the office, including your original attempts.
You may want to wait yet an additional few days after their return before trying to contact them. You don't want your final attempt of communication to be lost in a sea of emails that have piled on during their absence and, furthermore, it's generally polite to give them time to respond to your initial attempts at communicating with them.
Regardless, if so far they have not returned your requests, respond with one final email back to the responsible party. If that individual has a manager, be sure to CC them to the email. Once again, send this message as a response to the second email that you've sent. The CCed manager should see that you have attempted multiple times, over multiple weeks to further a discussion so that you can finish your work.
The ball is in the associated companies court. If they refuse to communicate, you can't make them do it. It is at this poin that I, personally, file away any associated documentation with the project and consider it "finished". You may be in the middle of "important" work but if those individuals who need your services refuse to communicate, you can't force them to talk. Instead, you can do your best to facilitate the needed conversations, document your work and, finally, forget about it if they refuse to cooperate.