The original Honeywell switch was a 2-pole switch. And typically with Honeywells, one pole is thermostatically controlled, and connects both blacks if it's too cold. The other side is a simple on/off switch that is regardless of temperature; both reds are connected unless the switch is in the OFF position. And the thermostat doesn't care whether it's switching 240V, 24V or millivolt. So toss it in your parts drawer; you never know.
Having a powered switch raises a lot of questions
Sanity-check that this is a legit product. It's a pushbutton (not a wind-up) timer, so it needs power to power itself. Normally, these devices use a neutral wire for that. Be cautious about ones that don't; If it has a UL, CSA or ETL listing with a 7-digit file number, then it's probably OK. Indeed you have linked the device in a comment, and it looks great. Clever GE!
Now we need to look at the heater's voltage. Absolutely nothing about your picture tells us whether your heater is 120V or 240V. The whites could be neutral, or they could be the other pole. This device is 120V-only and won't appreciate 240V none too much, since it will have heater voltage across it when the switch is off. The heater's nameplate will tell, or shoot us a pic of the breaker with a few other breakers around it.
We also need to check the unit's capacity. This unit has only a limited capacity of 800W. Good chance your heater is larger than that. This needs to be evaluated too - and nothing else will do but looking at the heater's nameplate rating.
You'll also need to identify which black is the supply hot wire; that goes to timer black. The other black goes to timer red.
All in all there are a lot of standing questions here.
Or just use a twist-knob type timer
Boom. All the above issues go away! Easy peasy.
The common twist-knob timer, requires only a 2-wire connection -- those same 2 wires already connected to the existing thermostat. The energy source for the timing mechanism is you. Its capacity is 2400W, we don't need to care about whether this is 120V or 240V, nor which leg is hot. Done and dusted.
Honestly, that's the way I'd go, unless there's some showstopper reason why a pushbutton type is required.
These types are available in a wide spread of time intervals between 5 minutes and 12 hours. A "Hold" position (stay on indefinitely) is an option. Obviously you want to avoid that option. I caution you to use one with a long enough time, so children don't have to set it twice. It's possible to trick these into never shutting off, so don't give them a reason!
This. This is why I hate pushbutton types. Having the switch twist the knob for you means the switch must be powered, and getting power to a switch raises issues that DIY installers aren't ready or interested in dealing with. A pox on all of them, the switches not the installers. Humans have wrists, they can twist their own knob.