A 16mth old will not "learn" mummy comes back because a child doesn't get the concept of time until around 3, as i understand it The 'fun' of peekaboo is the child learning that the parent comes back. The child doesn't need a concept of time for them to learn mummy goes away, and comes back.
No, the child may not benefit in terms of socialisation. What they will benefit from is a non-burnt out mother, which is essential.
Someone said "that is not how attachment works" but actually it is, with a 16mo, afaik. It's not. One day a week of rest a recuperation for the mother, while a child is in good quality childcare and reassured mum is coming back, is not going to 'break' an attachment built up over 16 months. Attachment is more like elastic than glass.
What OP is describing is signs of a secure attachment, crying when she leaves, settling when she comes back. She's not going to 'break' that by having a day a week to work and/or rest. One person mentioned 'good for her socialisation', along with the prime benefit here of rest for the parent. It may well be good for her socialisation. This isn't the thread to debunk any ideas that 16 month olds 'should' be in nursery. It's about managing transition to nursery for a child whose parent 'really really' needs the time. Jumping on one part of one post to criticise OP and wrongly say it will affect attachment is wrong, IMO.
OP says she needs the time, and is, quite normally, fretting over it, as we all do, and needs reassurance. It's ok for her to need that time, whether or not your assumption of "I am assuming that you are doing this because you have to, OP, you need to go back to work financially?" is true. Even if OP needs this time to drink tea and watch telly, she's not going to break an attachment built up over 16 months of this child's life by taking a day per week.