Doing the A levels won't help you, because most grad entry programmes don't take A levels into account. A few ask for A level chem, Barts want BB at AS, and Cambridge want more but they also want excellent grades.
I would recommend doing some research about the GAMSAT. I was in a similar position to you last year - my ukcat wasn't good enough but I still wanted to apply for medicine. I did some research into the GAMSAT and started preparing in January. I recently got my results, and my score is high enough to get an interview at St George's, and although Nottingham and Swansea haven't responded yet, their cut offs have never been higher than what I got. This is after I've sat the ukcat FOUR TIMES. I have a humanities degree.
The gamsat is really difficult, expensive, time consuming, and stressful, but if you have the patience for it, it is doable and it opens up a lot more options than the ukcat.
I hadn't done sciences since GCSE so I started really early so I could start learning chemistry, and also because I have a full time graduate job. You'll be a better position if you have A level chemistry, because most of the test is chemistry. Your psychology won't help that much, but it probably means you know a bit of biology, statistics, and you can interpret data / graphs - more useful skills.
I'd recommend getting an ebook called the Griffiths review, which suggests ways of preparing for it.
You can also sit it in March if you think you can get up to speed by then. Sitting in March means you'll have your grade before you apply.