Let me just start by saying that I start almost every morning with a nice cold shower. and although you don't need to do as I do, you will miss out on some of the amazing benefits of a cold shower
alertness
strenghtens immune system
heart rate increase
stimulates deep breathing (lowering CO2 levels in the body)
a cold shower is ideal because it activates the sympathetic nervous system, which is responsible for fight and flight. As long as you don't panic and start running you'll be fine. Cold shower helps you become more alert and clear. In studies it has been shown that the body eventually habituates to cold exposure and the parasympathetic nervous system takes over soon after - which is responsible for rest and digestion.
But look I'm not saying you 'need' to take a cold shower, it's not necessary, but it helps. A cool shower is just as good, as long as it gets you out of bed and moving. Even a warm shower... "What about hot shower?" Well I'm against that because hot showers should be saved for the night, since the hotness will active the parasympathetic nervous system (responsible for rest and digestion), and hot showers tend to dry out your skin and cause lots of other problems.
I don't just do it for the effects, but I also do it for the hygiene. A nice shower in the morning will make you feel fresh and clean. If you wake up and start your day without a shower you will feel like a filthy animal... "but TK am I not already a filthy animal?" well yes that s what you are, but you might as well 'pretend' and make yourself feel less like a filthy animal.
There's also this thing called momentum, it's starts early in the day, and if you miss it, you can't get on another "momentum train" chu chu. Showering in the morning will give you momentum because you started the day by overcoming some challenge. so why not overcome a few more, and a few more after that, you did one thing well, so maybe you can continue, go for a walk, do a bit of stretching, eat some breakfast...
I would recommend that after waking up, go straight into the shower and let the water run over you for a minute or two so that way you wake up and can start your day with some momentum. Not taking a shower in the morning will give it a lazy start and that will most likely follow you throughout the day. It's such a simple thing you can implement, so why not just do it?
Even Elon Musk says it's important to shower in the morning!
Sympathetic nervous system
https://en.m.wikipedia.org/wiki/Sympathetic_nervous_system
Parasympathetic nervous system
https://en.m.wikipedia.org/wiki/Parasympathetic_nervous_system
Cold shower versus hot shower which one is better?
https://www.healthline.com/health/cold-shower-vs-hot-shower#So,-which-type-is-better?