Frankly speaking, there is no need for you to buy the car. You should just take public transport everyday and put your baby at a CC close to your workplace. But since you already bought, here are my suggestions:
1) As many pointed out, it is best to put baby at a CC as close to your home as possible. Imagine you are on leave/MC? Imagine you change job and the continuity is affected? Jurong East is no go. Considering your work timing, I would say putting somewhere in CBD is fine.
- If your wife is driving, just drop you and baby off in CBD. This would also relieve her the stress for driving "alone" with the baby onboard.
- If you are driving, just drop her off in CBD or along any MRT station along the way for her to continue taking public transport. Overall travel time should still be faster for her. MRT is much more reliable nowadays, but imagine if NE line break-down? The time saved would be even more!
- You can do likewise when going back, have your wife meet you in CBD. But frankly speaking, when going back, it would be much faster for her to take MRT all the way back. But then since you already have a car, you can consider meeting her along the way from JE to CBD for dinner and drive back home too.
- Don't ferry your wife to JE and back in the morning. A lot of time is being wasted there, especially during traffic jam. And it is uneconomical as well with the fuel wasted. I think it is okay to ferry her back from work though. As said earlier, you all can explore various eatery after work! Enjoy the lifestyle with a car!
2) Negotiate with your boss for flexi timing. Not sure on your official working hours, but lets say you come earlier at maybe 8 work till 5/5.30 + 0.5~1 hr "showface" OT, you can still be in time to fetch your infant back.
- It does not matter if your wife is not on same flexi-timing. This mean she just have to take public transport to work and back. This can be a blessing in disguise as she would have flexibility in travel time (no need to follow fixed schedule, can be early or late sometime) and work time (unexpected OT).
3) There are still some cheap parking in CBD, albeit with some walking distance. Look around some more.
4) The best long-term solution is for wife to drive, and drop you and baby off in CBD. Going to work should be fine. When back from work, if she need to OT, just her be. You just fetch the baby and take MRT back.
Not advisable to shift house to JE. What if you all change jobs? It is okay to shift house to somewhere more "central" to maintain the flexibility though (like tiong bahru, commonwealth, lavendar etc).