Hello and welcome to BW forums
At 10 wks your LO likely needs a 3hr E routine, the A time would be around 1hr 15 to 1hr 30 but we can look at that and adapt the timings to his individual needs.
Could you spend the next 2 - 3 days observing and recording your LOs habits, this is how Tracy advised starting EASY and then using those timings to work out a suitable routine. There's a link here for how to start EASY, it has lots of information and links to various bits of info for people just starting out:
https://babywhispererforums.com/index.php?topic=186622.0Waking at 45 mins into his nap could well be that he has just not learned yet how to transition from one sleep cycle to the next on his own. This is part of the sleep training process. Getting LO down in the cot for the start of the nap and teaching him to sleep there helps, but LO might also need to be taught to nap for longer and be comfortable and confident enough to do so without fully waking at the end of each sleep cycle.
If you can record and post your observations (you can continue with whatever times you are used to or having been doing recently for this observation period) something like this including the exact times things happen and a brief note of what happened:
WU (wake up) 7.00
E 7.00
A 1hr 15
S 8.15 - 9.00 (45 min) in cot. tried to resettle for 30 mins, would not settle
A
E 10.00
A 1hr 30
S 10.30 - 11.15 (45 min) in cot. Tried to resettle, gave up after 20 min, slept in arms
S 11.35 - 12.35 (1hr) in arms
A
E 1.00
...and so on through the day to include
BT (bed time)
DF (dream feed if you do one)
NF (night feed)
NW (night waking not for a feed)
and up to WU next morning
I could then have a look at what is happening at the moment and suggest what I think might be a suitable routine to try.
Could you also let me know if you are sleep training at the same time as moving to the EASY routine and if so at what point you are putting LO into the cot and how this is going?
With regards to the E times, you say he is not finishing his bottles. It is advised that LO is given enough milk so that he can take his fill and there be an oz or so left in the bottle. If bottles are drained it is time to offer larger milk feeds. If however he is not taking much of his feed there could be another reason, please record how much milk he takes and at what intervals if you are unsure if he needs shorter or longer E times.
hope this helps. Please feel free to ask any questions you have.