OP,
I'm totally with you on this issue.
You are definitely not being unreasonable.
In fact, that you're even canvassing opinion on here, and have tried to let go a few other concerns about your MIL not respecting your (entirely rational) requests (and that you're trying to maintain family harmony, not out of a need for childcare, but to MIL/ DGS enjoy special time together), surely demonstrates you're the exact opposite?
Hope your little boy isn't too uncomfortable, that your soothing methods help him ASAP, and that you, your DH and MIL manage to resolve the situation re his care on her watch.
As PPs have said, a one-off, unfortunate accident by someone (usually sensible) truly not appreciating how fragile your DS's skin is, and who firmly ensures it will never happen again, is one thing...if, though, someone has disregarded your warning he will burn, and has since shown no remorse or assurances of more vigilance in future...well, that's far more concerning.
Seem to be a few MNers refusing to take seriously how very quickly and easily we sun sensitive, fair, prone to burning skin-types catch damaging rays.
(Pretty arrogant, IMO).
We're very happy you have the luxury of never needing to worry unduly.
"Oh FFS, it's only the UK, early May, hardly the Tropics" (usually followed up with "and it's cloudy, anyway") seems a prevalent attitude in RL.
Point being, it's how sun exposure affects the individual. Especially with babies/ young kids.
Not just you, or those known to you.
Yes, it genuinely is possible for some of us/ our kids to burn in what may sound laughably low levels of UV (which, btw, still penetrate cloud cover considerably).
Would've hoped it was widely accepted in 2017 that each childhood burning incident cumulatively increases melanoma risk in later life, for vulnerable demographics. Not in any way laughable. 
As 2 pale and freckled skin adult friends, whose (darker-skinned) parents were frequently negligent about sun protection throughout their early years, have recently discovered.
"A bit of redness will soon turn into a nice healthy tan", was trotted out often, apparently.
Their oncologists would disagree.
The cynics may want to brush up on your medical facts at the CancerResearch or SkinCancerFoundation official sites.