TBH, I always struggled with being away from home. Seriously struggled with it. I remember being put over the fence aged about 4 or 5 to go and play at my cousin's neighbour's house without any adults I knew there and I hated it and kept clawing at my cousin saying "Give me a kiss, give me a cuddle!" I couldn't get through a sleepover without feeling sick and frightened and crying and needing to be picked up, until I was about 14 and even then it was only OK when I was with a friend whose house I was extremely familiar with and had been to lots and I knew their parents etc. I was also in theory fairly used to staying away from home as my parents were divorced so I spent EOW at my Dad's house since about 6 - but that was like being at my second home, so never really counted quite the same.
Went on French Exchange when I was 15 and spent the week basically inconsolable and totally unable to cope. Was better during the day when out and about and distracted but once I was on my own in my room or having dinner with the family etc I couldn't eat or relax and I felt like I was choking all of the time. I used to hate it even if my mum went to bed before we did and I would lie awake feeling extremely anxious.
Even aged 16, with college, we went on a trip to Amsterdam and I spent the whole first night hiding in the hostel room crying! On the second night, I was persuaded by somebody to go to the bar, and apparently alcohol was the cure to my homesickness
- never had a problem, actually, since that day.
I moved out of home at 18 and moved abroad aged 24 and I never feel homesick any more. I think it's totally fine to be a late starter and even if you do have anxiety about being away from home when you're younger it can be easy when you're older, it's just a case of maturing.