Are you going back to work at all? That will help a lot even if you only get to read/listen to a little bit of music on your commute and/or talk to colleagues about non-baby stuff during the day.
As for your specific questions (haven't RTFT yet so apologies if any of this has been covered) :
For context, I have a 1-year-old (just had his birthday this week).
1. Have a shower, blow dry your hair and put make up on with them in your care?
TBH I have always been able to do this. Either because my husband looks after our little boy while I do it, or because (during the working week) I get up before everyone else to do it, or because (in the school holidays) I remove anything dangerous/that I don't want damaged from the bathroom and just let him roam free while I shower. I can see him, he can see me, and he's not bothered by the noise of the hairdryer (helps that I don't wash/dry my hair every day...speeds things up a bit!). He also enjoys playing in/exploring the bathroom as it's a different environment for him (even if this means I have to fish stuff out of the bin afterwards that shouldn't be in there
).
2. Regularly exercise?
I go straight after work to swimming/the gym 4 days a week (though not lately as I have been ill) and then on one weekday evening my husband comes home a little early from work so I can go to Pilates. I try to only exercise during the week so that weekends are kept for family time. But if my working hours didn't permit it, I'm sure my going to the gym Saturday/Sunday mornings as before wouldn't be a problem either as my husband enjoys spending time with our little one (obviously) and he knows it's important to me.
3. Eat a meal whilst they're awake without rushing it as you can't really spare the time?
Again this has never really been an issue. We have dinner after he has gone to bed. Breakfast is always all together and was even before I returned to work. We get up early enough so we don't have to rush it (little one is usually awake between 6 and 6.30). As for lunch...either I'm eating it at work or, if it's holiday time, LO and I eat at the same time (at about 12) and he plays on the floor while I finish.
4. Watch a TV show you want to watch whilst they play independently?
When I was on maternity leave I'd watch TV while he slept. Now he's older it's TV only when he's gone to bed, or (if I want to have something on in the daytime) it's something quite mindless that I don't have to concentrate too hard on.
I just refuse to believe that parenting is all doom and gloom and that you cease to become a person when you become a parent.