I am a paediatrician (a doctor that looks after children and babies).
People are often most concerned about the MMR because it is the vaccine that was targeted by Andrew Wakefield. He was paid to discredit the MMR vaccine by solicitors interested in making cash off of claims that it was harmful and linked to autism. At the same time he patented an alternative single vaccine against measles. In order to meet his own financial interests, during the infamous MMR-autism study he conducted, amongst other things (fraudulence etc) he subjected the autistic children in his study to painful invasive procedures they did not need - colonoscopy and lumbar puncture. He doesn't care about children. He hurt children for money. I don't think it is said enough.
Though fatality from measles is very rare, the illness is no joke. And if a child is unlucky (around 1/1000 mortality) and does get a serious or fatal complication such as encephalitis, sepsis, SSPE and so on with measles, there isn't a treatment that can stop it. You can't take back or make up for a decision not to protect your child. Rubella poses an appalling risk to pregnant women and mumps can cause infertility. All of these illnesses are seen in children right now, in this country, because of low vaccine uptake and for no other reason. They are becoming more common. The risk is getting higher and higher. Unfortunately, I think vaccine refusal is not going to abate until measles is so common that many children die and people become truly afraid. People are never afraid soon enough. Beat the curve!
Speaking for myself I am so concerned about exposure of my own baby to measles through my work and in general, that I decided to request an extra MMR at 6 months old. The relationship between MMR and autism has been investigated thoroughly and repeatedly. It doesn't exist. My son found the immunisation painless but was grottier for a couple of weeks a short while after. Related? I don't know. He is his perfectly, happy, cheerful healthy self now and I will make sure he finishes his schedule. I pursued the extra immunisation because however unlikely it is, I could not entertain the possibility of him suffering with a serious illness because I failed to protect him.
If you do not feel comfortable about your baby having MMR, the single immunisations are as effective. Please, do whatever you must to protect your child from these awful diseases.
However, I would advise that if you go for the separate injections, you will have spent money you could have saved for something else, and you are definitely or potentially increasing the risk of actual side effects of immunisations (sore injection site, risk of fever, feeling unwell, rare instances of things like ITP etc) by having three times as many separate immunisations.