Either is OK.
"Please come to cardiology clinic in time as mentioned in appointment given to you."
My understanding from this is to be punctual. So, I think it should be on time, shouldn’t it, please?
Either is OK.
Please come to the cardiology department on time = Please arrive at the time stated on your appointment card.
Please come to the cardiology department in time [for your appointment] = Even if you rush in with 2 seconds to spare before your appointment time, you must get here before (or just before) your appointment.
There's not much difference but it's there. Note that I have added "the" before "cardiology department". If you drop "department" then you can also drop the article: "Please come to cardiology on time". However, most hospitals don't use this abbreviated form when addressing patients. They tend to use it between hospital staff "Dr Smith to Cardiology stat" = "Can Dr Smith please come to the cardiology department immediately".
Remember - if you don't use correct capitalisation, punctuation and spacing, anything you write will be incorrect.
A rather pernicious development has recently begun to appear in Canada (and I wouldn't be surprised in the US as well). They give you a little card that states the time of your appointment and tells you to arrive ten minutes before that time. So not only do they never see you on time, now they are adding an additional ten minutes to the time they keep you waiting. It's like a little service compris charge on a French restaurant bill.
In the NHS (UK health system) it's not formalized like that, but it's understood in some cases. For example, if you have an appointment with a consultant whose assessment will depend on the results of a blood test, you'd better arrive 10-15 minutes early!
If you're required to go for a blood test before your consultation, you will either have two appointments (one at the blood test clinic and one with the consultant) or an appointment time with the consultant but with the annotation "You will need to attend for a blood test 30 minutes before your appointment". If the NHS tried to rely on people simply working out for themselves that they need to come early for a blood test, the whole system would fall apart. Oh, hang on, it is falling apart!
Maybe there is such a printed warning, but I haven't seen it. There certainly aren't two appointments (in the sense of an agreement to see someone at an agreed time). When I attend such a clinic (which I've done about 20 times since 1995), I get one appointment. Patients arrange their own blood tests, as if queuing at the deli counter in a supermarket (with exactly the same mechanism - taking a chit and watching a digital counter). (And you're right about the system falling apart - for my first appointment, I was ignored for two hours because I didn't join the phlebotomy queue - like all the old hands. )