Hi, i had similar issue when nap transitioning.
i did so much research i could write a book and i even paid for a sleep schedule - this is what worked for me and i will do again with dc2 when born (if needed)
power naps are your friend. it will take a few days for change to happen so you really have to be consistent. at 15months could be anywhere 3-7days but as we always get told all children are different so some get it quicker and some longer but stick with it.
you say he can only last until around 10.30 for his nap where really you want his one nap to be after lunch time (around mid day- ish).
so for power nap either put him down in cot for nap, or let him sleep having a cuddle what ever works for you - i'd let mines sleep on me as it was easier to monitor how long she was asleep for.
do not let him sleep longer than 15mins - yes i know this sounds incredibly short and yes it may be a battle to rouse him again and yes he may be cranky but the aim of this nap is to keep him going until his real nap time (the time you want it at).
so i would recommend trying for power nap at 10am (before he gets really tired) so that he has fallen asleep and woken again before 10.30 - ideal awake by 10.20 or early the better really - so that he's build up the need to sleep again by between 12-1pm
as soon as he wakes offer him a drink and light snack and then distract distract distract to start with he may just want to sit on you but interact with him to try wake him up fully.
see how he goes and aim to put him down for nap between 12-12.30 so offer lunch between 11.30-12
if you take this advice you might find that the power nap for 15mins means he can last til 1pm before nap or you could find that he wants to then nap much later.
its will be a bit of trial and error to start with to see what works for you.
at one point i was doing a 5min nap at around 10.00 with mines as if i let her sleep longer when she woke she was full of beans for ages.
once wake up occurs later and bedtime is later and your happy with that drop the power nap.
while transitioning the times of the nap/dropping the power nap it is normal for dc to be cranky and sleep a bit like they have jet lag this is normal and passes quickly, you are litterally retraining their bodyclock.
i remember starting it and on day 2 my dc was a moany crank and i felt like i was making her miserable and was ready to accept the early (middle of the night) start to the day but kept saying i'll give it one more day, then another and by day 4 mornings started later and bedtime was getting later. so was defo worth spending a few days at home so she didnt fall asleep in car or buggy etc
my little one wakes now at 17months between 7.30-8 am and this works brill for me as i can get up and showered etc but she wakes.
sorry for the long msg - hope either this helps or you find something that works as early starts are a killer