Hey Barb, I'm an American who has been contracting in programming/ Databases for the last 15 years. I've had many different periods during that time when I was looking for a role and NONE of them were less than 2 or 3 months. Most UK companies would struggle to organise a picnic in a month, much less hire someone. Don't sweat it until you reach 6 months with no interviews. Maybe more if you live way up north. It's grim up there.
I doubt anything is wrong with you or your CV, agents will plainly tell you if there is . However, I'll be happy to look over your CV if you like, and send you mine. PM me. If the agents are all telling you something, like you need to increase the area you are looking in and face a long commute, you should listen.
On a practical note, I would find the four biggest job sites and every day go to a new one and re-upload your CV, as well as apply for any advertised positions. Each time you upload your CV, it is distributed to all the agents. Contrary to what they have told you, agents don't "keep anyone in mind", they tend to send out the last CV that was on their desk. It's all a numbers game, send out as many CVs and talk to as many agents as you can. The first agents that call are just filling their databases, so if you don't hear back from them it doesn't mean anything. The guys who have the jobs will call later, eventually.
I might consider going to a training site like Pluralsight and studying up on some of the skills that you frequently see on jobs you are interested in. Even if you just spend 2 days taking a web course, put it on your CV. Key words are everything! I personally don't think anyone cares about qualifications for project managers or programmers, but you've got to be able to convince people you know what you're on about by throwing around the key words. Also, you must have the appropriate key words all over your CV so you come up in a search.
Finally, if your only "issue" is being American, I think you benefit from that. When it comes to tech, they love Americans. Do you have anything else going on? Pregnant? Girl Power facial tattoos?