Replaced 4-way electrical switch with new one with ground. Will it work without the ground?
I hooked up the 4 wires in the same position

That's your problem right there.  Common blunder with 3/4 way switches.
Manufacturers completely randomize screw positions on the switches.  Every switch is different.  You cannot rely on positions at all.
You must use screw colors.
Fortunately, this is not too bad to clean up: simply group your 4 wires by the cables they come in on, so you have cable 1/wire 1-2...... and cable 2/wire 1-2.
Now, on cable 1, put its 2 wires on the black screws.
On cable 2, put its 2 wires on the brass screws.
And you're done.
On your switch grounding, if the junction box is metal, your work is done.  Switches ground though the 2 mounting screws.   Otherwise if you see ground wires in the back of the box, you can pigtail a ground wire off of them and bring it to the switch.  Note that adding a pigtail can entail a number of supplies and tools you don't necessarily have!
Lastly, a 1-second delay in a light starting cannot be caused by any plain switch.  Unless your existing switch(es) have some sort of smart-switch/dimmer feature, it ain't them.
A 1-second delay is typical of certain fluorescent and LED lights, however.   In the case of fluorescents, it is the tube being pre-heated for a soft start that extends the life of the tube (which is VERY long).  In LED, there's really no reason for it.