Delay of boiler firing?

It depends on how the system and boiler is being controlled. Generally there is a work around but it depends on how involved you want to get and whether its worth the investment.

seems like an aquastat on the wood boiler that opens t-t to the gas boiler would be the way to go.

Why does the gas boiler need to have a call for heat at all if the outdoor wood boiler is already firing?   Are you firing the wood boiler without dissipating the heat somewhere?  Your query seems to assume that there is a thermostat that is calling for heat on a zone (or the entire system) indicating that the call for heat is somehow off at times.   Now that begs the question that you have a hand fired boiler that is generating heat with a circulator pump to a HX that is somehow off.  That wood boiler will then be generating heat with no way to get rid of that heat.  That sounds wasteful to fuel a hand fired boiler and not use that heat that is generated by the fuel.

I would propose that you learn to feed the wood boiler in such a way where any heat generated will be put into the home or the zone as it is generated.  That way you will have a hotter home and adapt the amount of fuel you feed to the wood boiler based on how much or how little the home or zone overheats.   That is how the folks used hand fired boilers in their basements before oil heat and gas heat became the norm.  After 1920 you might live in a home where the winter indoor temperature was 85° inside with the windows open.  (read about the Spanish Flu and how it affected heating system design)  Even before that in the 1800s coal fired boilers in homes were hand fired and the only regulator was the dampers that controlled how much air flowed thru the coal fire, and how much coal you shoveled into the fire box.  

However if there is a system that is hand fired, and you are using that heat elsewhere, and you want to draw some extra heat to heat a zone based on a thermostatic input from that room, there can be a circuit that can be designed to accommodate the room temperature thermostat calling for a circulator to operate to draw heat from the Wood Stove and preventing the gas from firing on the boiler if there is already enough heat available from the boiler even though it may take some time to reach the rooms. 

If there is a need to have one thermostat in that room, then I can design a logical set of relays and temperature sensors to accomplish your needs. Please explain what components are in the system and how they are interconnected with the piping from the wood boiler, to the pumps, and the heat exchanger, and any other system accessories.  For example: is the wood boiler’s heat being used any place other than the home or zone you want to heat in the original query, or is the wood boiler only to heat that zone or home?

…unless you meant "second hand #2 heating oil boiler", or "previously owned", or a "regular oil boiler that was used in a different location".

I may have jumped to a conclusion about your term: USED OIL boiler and you really ment …used OIL BOILER. Either way, that control system is still easy to design. What is the oil boiler make and model number?

I usually put a strap-on aquastat on the outdoor boiler supply line and use it to interrupt the call to the gas boiler. As long as it senses that the outdoor boiler is hot, the gas won't fire. You may need a switching relay as well if you don't already have one- the gas boiler would not run the pump(s) without a thermostat call.

Without the actual model numbers of your actual equipment, i have used some generic wiring devices to illustrate what I believe you have existing, although I do not know what is making the oil boiler call for heat to operate the burner and circulator pump.

A WiFi thermostat is operating a typical ModCon-Combi boiler's Central Heat (CH) circulator pump and the gas burner. It was installed as the manufacturer recommended.  After that system was installed and operating properly, there was an auxiliary outdoor oil fired boiler added with a heat exchanger so that the antifreeze water and the indoor CH system water were separate systems.  

I do not know what might make the call for heat to the oil fired boiler might be since the room thermostat is operating the CH by way of the Combi boiler.  That is why there is a question mark (?) on the diagram where the oil burner control thermostat is supposed to be connected.   

If you can provide the model number of your Oil boiler, your Combi boiler, and your thermostat, I can draw you a diagram of the best way to operate your system.

I looked at the VESTA Combi Boiler instructions. It appears that there is no room thermostat with the standard wiring of this boiler. Since you indicated that you have a Honeywell WiFi thermostat, I was interested to know where it is connected to, on your boiler.

Also, does the oil boiler have a control that a thermostat might be connected to that also operates the circulator pump? Perhaps a Honeywell L8148 aquastat. If not then how does the burner know when to stop operating at 160°, and how does the circulator know when to move the water from the boiler to the heat exchanger? Is the oil boiler circulator always operating constantly?

A cold start oil boiler will use less fuel, or perhaps one that is set at 80° and a call for heat will ramp it up to 160° will keep the boiler warm on the off cycle but not use as much fuel.

I believe that the Beckett AFG burner is equipped with a primary control that has a manual reset button for flame failure. That control may have a thermostat terminals labeled T T. Those are not the thermostat terminals I would like to use. I would like to use the thermostat connection that will operate both the burner and the circulator pump.    

The strap on aquastat that @GroundUp suggested seems to be an easy fix. A "C" aquastat could energize a relay to close the gas boiler thermostat contacts on temperature drop.

So you have a low temperature radiant floor system that is properly installed according to the VESTA manual. That should have a high temperature of about 120° to 125° to operate properly. The OIL boiler will operate at a minimum of 140° and stay at or above that temperature based on what control you have, because that is cast iron and you don't want to experience flue gas condensation on a regular basis. That makes sense. Now you need the HX to convert that 140° or higher temperature oil boiler water to 120° for the space heating part.

This is easy with a inexpensive mixing valve when it is time to connect the plumbing to the outdoor oil boiler & HX. The wiring will still be similar, you may need 2 circulator pumps to move the heat from the oil boiler to the HX and from the HX to the system. I'm going to suggest a piping arrangement but @hot_rod Bob Rohr is the expert on the piping side. He will keep in mind that your are not interested in adding expensive hydronic components. He can design the best piping connections.

Can you let us know what piping components you already have? The Boiler obviously, and perhaps the properly sized HX. Do you already have any circulators or mixing valves that you plan on using to connect the "free fuel" waste oil burning system to the radiant floor system. Also sizing is important. So the model number and size of the HX and any other hydronic accessories you have will be helpful.

Here is a basic piping design that will get you where you want to be with the lowest cost components. Bob may have improvements. The wiring design still looks good for this arrangement. I just need to bring on the HX pump with the outdoor boiler call for heat. The outdoor boiler pump will operate whenever the oil boiler is firing so the water circulates to the HX which should be located inside the Vesta gas boiler room.

The blue shaded area is the oil boiler additional piping added to the Vesta boiler's I/O manual drawing for a one zone system …where I added the radiant heat emitters and removed the baseboard emitters.

EDIT: I might make a note here that both boilers will not run at the same time. This piping is for an either or scenario. The Gas boiler will come on when the oil boiler fails to operate by the second stage of the thermostat. That will usually be when you run out of fuel or if there is a flame failure condition with the outdoor waste oil boiler. The Combi boiler will operate as a backup. I would change the system connection for the HX if both boilers were designed to operate at the same time in order to maximize the output of both boilers.

Selecting a Flat Plat heat exchanger can be challenging. @hot_rod can usually help in this area.  Knowing that the oil fired boiler will be producing a minimum of 140° with a high temperature of at least 160° is necessary in the calculation.  The amount of heat needed to heat your space is also needed. Does your 80,000 BTU Vesta boiler have any problem heating the home when it is very cold outside?  The flow rate through the HX on both sides are needed to exchange the proper amount of heat.  Bob will have some questions for you when he checks in.

Bob may have a better design for connecting the HX to the existing boiler system. He needs to know that it is properly connected and designed. I would guess that the original installed used a piping design close to something like this.