In order to get a better idea of what the children went through, here is some medical information gathered from Medicinenet.com, edited a bit by me:
"Measles is among the most readily transmitted of all infectious diseases, similar to chickenpox in this respect.
Typical measles begins with fever, runny nose, hacking cough, and red eyes. The very characteristic spots within the mouth appear 2 to 4 days later, are often on the inside of the cheeks, and look like little grains of white sand surrounded by a red ring. Sore throat occurs along with inflammation of the airways.
The rash appears 3 to 5 days after the onset of symptoms. The rash progresses from the head downward. It begins below the ears and on the side of the neck as small irregular bumps that soon increase in size and spread rapidly (within 1-2 days) to the trunk and limbs as they begin to fade on the face. Bleeding spots and bruises can occur with very severe rashes.
The fever can top 104° F (40° C). The eyes are reddened and watery, very sensitive to light and there is swelling around the orbits. There is a hacking cough.
The course of the disease usually follows the course of the rash. In 3 to 5 days, the rash begins to fade, the fever falls abruptly, and the patient feels more comfortable. At this point, the patient now can no longer pass on the disease. The rash may leave discoloration in its wake and the discolored skin may peel.
There are a number of different types of complications, some of them very serious, even fatal, that can and do occur with measles:
Bacterial infections: Pneumonia, ear infections, and other bacterial infections are common complications. People with measles are vulnerable to strep infections. Measles can reactivate and worsen tuberculosis.
Acute thrombocytopenic purpura: Low blood platelet levels (important blood clotting elements) with severe bleeding constitute a potentially serious complication during the acute phase of measles.
Encephalitis: Brain inflammation (encephalitis) occurs in 1 in 1,000 cases. It starts (up to 3 weeks) after onset of the rash and presents with high fever, convulsions, and coma. It may run a blessedly short course with full recovery within a week. Or it may eventuate in central nervous system impairment or death.
Subacute sclerosing panencephalitis (SSPE): The measles virus causes subacute sclerosing panencephalitis (SSPE), a chronic brain disease of children and adolescents that occurs months to often years after an attack of measles, causing convulsions, motor abnormalities, mental retardation and, usually, death."
As you can see from reading the above description, this was (and still is) a dangerous disease with potentially fatal complications. I have read that the disease was introduced into the palace by a playmate of Alexei's. Of course, it would have spread rapidly in the confined atmosphere of the palace. The children certainly experienced the symptoms described above, such as the rashes, fever, sensitivity to light (remember they were kept in darkened rooms), cough, etc. In one of her letters Maria mentions that Olga "has become voiceless because of the bad cough." And they also experienced complications such as ear infection and I don't recall if pneumonia was suffered by one of them as well. Again, Maria writes "Tatiana has got pain in the ear therefore her head is covered with bandages and she can't hear anything." Interesting to read about the potential complication with the low blood platelet levels and how that would have affected Alexei with his already compromised clotting function. He must have been especially vulnerable during this illness.