Congratulations on your new job. You get to be the first nonparental caregiver in this little boy's life. Good for both of you.
I'm of the belief that kids do well when they get to experience a nonparent caregiver who cares for them, is trustworthy and responsive and loving and predictable.
When I was a nanny and starting with a new family, or when I had new children in care (preschool and toddler groups), my first goal was to establish a trusting relationship with the child, no matter what age. This will mean creating a daily routine, so that days are predictable for this little guy. Sometimes, this means packing them around, lots of loving touches. He sounds very scared and insecure, so be sure to stay close, tell him where you are going if you leave the room, and be predictable in how you disclipline. (at this age, it will likely be distraction, redirection and very simple either/or choices) Love on him as much as you can. Gentle physical play which allows both kids to crawl on you is great. Fingerplays and puppets for stories will give him something else to focus on, even momentarily.
Some people will call it coddling, but at his young age, he needs to attach to you if he's going to come to enjoy being at your home. Make yourself emotionally available and try to use positive language to provide empathy and acknowledgment of his sadness. "I see you are so sad because you want Mama. I know." and then, try a redirection of that "Let's write mommy a letter. 'Dear Mommy, I will be so happy to see you. I love you.' (writing notes-- this worked VERY well with my toddler groups, by the way) Now, do you know what I was thinking about today? I was thinking about boats! I'm going to fill up this pot with water and what could be a boat? ..."
Okay, all this last part looked a little goofy, but that was how I would often get little kids to shift their focus. Acknowledge the emotion, address it to the proper party (the parent , via 'letter', which is about you articulating how he's feeling, this offers empathy and validation and helps you to bond with him) and then moving on to some exploratory play by asking a simple question. You'll find what works for you from this. Make bonding your goal and don't be surprised that he's not engaging with your son; kids generally don't play 'together' until they are older, into their twos and threes. Most of my toddler group (20-30 months) were mostly engaged in parallel play and watching each other; interactions were mostly around toys. So he sounds fine for his age.
How long it will take him to become comfortable will depend in part on your receptiveness/responsiveness and on his temperament, which is one part you just can't control. You are right, though, this is a huge adjustment for him, so keep your immediate expectations low for now. If you are providing loving support and offering things to do (such as more exploratory play) *with* him, he'll come round.