I don't think that would be such a big problem, actually.
No one is born with immunity to modern diseases. It is not really in our genes, apart from the non-specific immunity that is due to the cell membrane properties and inflammatory responses etc. - but a Neanderthal would have all that, I presume. We haven't changed all that much.
Microbes mutate so quickly that it would be impossible for our genes to keep up with the specific immunity to specific modern microbial strains. Luckily our genes have been programmed with the ability of the immune system to
learn to defend itself against the microbes it encounters. Therefore, IMO, a member of an ancient species who was born and brought up in modern times and encounters modern microbes would develop resistance to modern diseases, not ancient ones.
If a Neanderthal child would grow up normally with




Sapiens humans (as opposed to a sterile lab somewhere) his situation would be different from the Native Americans who encountered previously unknown microbial strains when the Europeans came. During his childhood the Neanderthal child would be exposed to the same microbes




Sapiens children are and I think he would develop immunity the same way they do.