Most Medical schools will require Chemistry A level and another Science. Grades are usually in the range of AAA - AAB.
There are other courses other than the standard 5 year degree. 1 is a six year degree course where you do a foundation year. This is for people with non scientific A levels. Alternatively, you could do a degree and then apply for a graduate medicine course (4 years). Some will allow any degree discipline, some will state specific subjects they will accept. Generally you need a 2.1 or above.
You will also have to sit an aptitude test (BMAT for Oxford, Cambridge, UCL and ICL) and the remaining universities require you to sit UKCAT.
So after your degree, you will do foundation year 1 and then 2 and then go on to your speciality training (in your case a paediatrician).
Of course, being a doctor is not just about being a medical textbook. They also look at personal qualities. At the end of the day you could have straight A*s at A level, but if you have a **** personality and have absolutely dire communication skills, the chances of anyone wanting to work with you within a fast paced environment is minimal. You also need work experience to support your application, preferably within a hospital/caring environment.