How will Saywer react when they meet their mother for the first time? How will both parties feel?
Saywer will feel... A lot of things.
Happy, but also sad, angry, betrayed.
Why did you not take me with you? Have you been wondering about me? Do you even care?
Anesa will approach them with nothing but warmth. They wanted to bring all her kids so bad, but she knew if she didn't at least leave one child behind Sirius would hunt her down and then all three of them might've been in danger (Although I may rewrite this and say that Sirius has Saywer with him at the time, and Anesa only had one chance to escape and couldn't miss it)
Anesa has missed her child so much and been thinking about him everyday, she'll hug them, give them a kiss and tell him how much she loves him.
They might not take it too well? They'll be overwhelmed for sure. This'll be not only the first time interacting with their mother but also the first positive and loving reaction from one of their parents (not parental figure, their actual parents).
At first they'll return the hug and bask in the affection given, but they're going to need a bit more space and some time to deal with their emotions. Anesa tries to be as close to them as possible, but eventually does give them the space they need even if it saddens her.
Saywer is understandably angry at her , out of all their siblings why were they left behind? She has no right answer for that, she just made a decision without much of a plan. He knows that if she could she would've taken them with her, but what's done is done. Now they're going to have to deal with a parent that unlike his other one treats him the exact opposite. They have a hard time distinguishing whether it's real love or if it's manipulative like Sirius's. He finally learned to let his siblings love and care for him (although they still have a love of doubts about it, those will go away slowly) but his mother is a different story.
Anesa approaches him with nothing but love and happiness, and although Saywer does enjoy said affection they're going to need time and space to deal with their emotions before they can take the first step towards building their relationship.
Thanks for the ask!! I love Saywer so much.