Two separate issues:
A single sector flight where the previous days flight is delayed. If you look at FR24 you will see that this happens quite often. Generally a suffix is added, 'D' 'Y', etc. ATC doesn't necessarily use the published flight number, BA domestic flights were (and may still be) something like Shuttle 1 Alpha where 1 is the route. Air Canada might add 2 to the start of a flight number in case of delays.
ATC will advise crews if similar numbers (Speedbird 93 / Speedbird 193, etc) are on the same frequency.
As an aside it is not unknown for two flights of the same number to be scheduled on the same day when the clocks change. For example a HKG -LHR flight might depart at 00:15 and 23:15 local time (Lets not worry about UTC / zulu time.) In that case one flight might might be given another such as BA27 / BA1027.
Second, there may also be flights with multiple sectors, sometimes with a change of gauge. This is far less common now but in the past was quite common. For example Pan Am might have had a 747 fly into LHR from JFK and then onto Hamburg as a 727. It was not unknown for the second sector to depart before the first had landed. I think that I a number was added to the start of the 727 sector, e.g. Clipper 4102 - though I'm not sure.
Right now in the US it is common for outbound and inbound segments of a rotation to have the same flight number. (Really confusing if you ask me.) These are generally operated by the same plane from a hub to an outstation so you would not expect there to be any issues but I dare say that a substitution has ocasionally been made and the inbound and outbound are airborne at the same time. US airlines often use 8000 / 9000 codes for positiioning and charter flights and one of those could be used.