The other children are "down here"; they are at this place, which is implied to be geographically "down" from where they were before. This probably means moving from an urban to rural area, going from the high city to the low countryside.
There is an implied "having" between "after" and "measles"; the phrase can be understood as "after having measles". "As likely as not" means, especially in British English, "very probably" (Collins Dictionary). Thus Martha thinks it is very probable that the other children have come "down" to this countryside after having measles.
The children may have been in a home for convalescence. I had some difficulty locating a good reference for this, thus both quotes are from the same source: Locating Convalescence in Victorian England. This article cites a bunch of sources. Go check those out for further reading! The time period covered is a little before the book's publication, but the same general principles would hold.
convalescent homes, as they were often called, began to appear in the middle of the nineteenth century... By the end of the century, more than three hundred convalescent homes had been established
"Convalescence" is the time period after the acute phase of illness, but before you are fully recovered. For measles, perhaps you no longer have the rash all over your body, but you are still fatigued every so often and cannot carry out normal activities. You are still sick, but in the recovery phase.
Crucially, since the children would be here after their acute phase of illness, it would make perfect sense to say they had come "after measles".
The designers of a convalescence home specifically for children explained why they were in the countryside thusly:
“a few weeks … spent under the invigorating influence of fresh-sea breezes — days passed in the open air, and free, happy exercise away from the confinement of streets and towns” would “disperse all the symptoms of their former existence”; a timely stay would “convert puny, wan, and sickly little creatures … into hearty, healthy, robust children.”
Working-class life in the city was hard, messy, dirty, loud - a terrible place to recover in. Thus the children would go "down" from the city to the country to convalesce, in a peaceful environment believed to be more beneficial to health.
They were also encouraged to exercise and play out-of-doors, which would explain why Martha would think the protagonists would be meeting them ("picking up"). Children running around outside are wont to meet each other.
It also makes sense that Martha would see such children as "strange". They would have come from away and will leave soon once they are better, thus, they are strangers to this area.