When I had elevated liver tests, my GP ran a battery of blood tests, and had me back in two weeks for repeat liver function tests. When they were still high, I was booked in for an ultrasound and a referral to the liver clinic at the hospital. I had the ultrasound within four weeks of the initial high reading and an appointment to see a consultant two weeks after that.
Granted, all of my liver function tests came back elevated, which may explain why I was seen so quickly, but my numbers started a little bit elevated (about 3 - 4 times what they should be), and had risen to severely elevated (about 20 times) within six weeks, at which time I started feeling very ill.
I was subsequently diagnosed with a fairly rare autoimmune liver disease, and I doubt that's what you have, but I would still try and push for a faster follow up if it's possible. There are ways of speeding things up - for example my GP had sent the referral to the liver clinic when she had the first results as it can take up to two months to get a referral.