"Separation Anxiety" is not something finite..... it comes and goes per each age-stage.
At this age, it's what happens. BUT, keep in mind that separation anxiety will come and go and will be manifested in different ways, per the child's age-set.
Your child is in unknown environments now... so, its a matter of adaption too, in conjunction with their emotional/cognitive development....
It's hard for the Parent, but also hard for the child... they are going through a doozy. They can't help it. They do NOT do it on purpose.
PLUS he was sick... and being ill naturally makes a child more clingy. SO... for him, there are MANY different things going on at the SAME time.
It's just something that each child adjusts to differently, and per their personality and disposition and sense of security and trust.
Yes, he will and is, having a meltdown when he anticipates you leaving him, and when some "stranger" is with him, even if they are relatives. I would not 'force' him to handle it... a child this young does NOT have "coping skills" to handle emotions that they do NOT understand.
It may be a 'vacation' for you... but it is not for him. He doesn't know what a 'vacation' is. All he knows is he is feeling insecure... and I doubt that in 3-weeks he will be 'cured.' And he probably knows his Mommy is stressed about it... which just makes it worse. Kids can feel vibes.
The thing is.... his sense of "consistency" and regularity and routine... is non-existent now. Thus, his anxiety. Then, in 3 weeks, he will fly back home. He will need to adjust again... and to jet-lag too. 3-weeks for a young baby/child is either too long, or just long enough to get used to things... but by then, it will change again and he will be back home.
It can't be helped now. It is what it is, and he needs you and the comfort that only a Mommy can provide. OR, if you can handle it...and you REALLY need a "break", then you leave him with Grandma and go have a break for yourself, knowing that he will cry. Sometimes, this can't be avoided either, depending on your circumstances. Likening this "separation" to a baby who has to go to Daycare...
I don't have a concrete answer for you... just some ideas and thoughts. But, be aware that "separation anxiety" is not just something that goes away, and it does return at different ages and phases and per the cognitive/emotional development of a child. Even 4 year old kids who go off to school get separation anxiety. So keep "expectations" in line with their age-stage. Just as much as it is hard for the Mom... it is equally or even harder, for the child.
My own son is now 2.5 years old, and although very self-reliant/independent.. he is going through a "clingy" phase with me and just LOVES to always be attached to me. Literally. He just always wants to be attached to me like an extra appendage and climbs up on me like a little baby Koala Bear and he'll just have this HUGE happy grin on his face and just keep repeating "My Mommy.... Love Mommy...." Its hard but so cute... and I can't get 'mad' at him for it. And he even gets "protective" over me... if my daughter or others come near me, he will "sometimes" say strongly "MY Mommy..." and he'll put his arms around me & shush others away! He can't help it. Nothing has changed in his environment or our family... but HE is changing. It's purely developmental. I know it's a phase though. So it will pass, but who knows when.
Keep in mind ALSO... that the "bonding" of a child with their Mommy ALSO changes and ebbs and flows. All developmental based and per the child's own individual self.
One day, our kids won't even want to be near us or with us....
All the best,