Integrating existing Hydronic Radiant with new Hybrid (dual fuel) Heat Pump (forced hot air)
What you're describing is a waste of perfectly good heat pump heat

Forced-air, "dual fuel" heat pump systems require a "hard cutover" from using heat-pump heat to using auxiliary heat as you can't run a DX (heat pump) coil and a furnace heat exchanger (HX) in the same air handler at the same time without playing havoc with the heat pump side of the system (due to the hot air from the furnace impinging on the heat pump indoor coil).  However, since you already have a hydronic system that can supply standby heat (either as radiant heat or by using a hydronic coil in the heat pump air handler), you don't need to make that compromise, as even with a hydronic coil in the air handler (vs. radiant heat), you can run it and the heat pump indoor coil at the same time, allowing you to maximize the amount of efficient heat your heat pump is giving you, all the way down to your system's balance point and even beyond.

"But", you might say, "that new furnace is 95% efficient! My old oil boiler could never match that!"  While you are right that your old oil-burner can't condense the water out of its flue gasses, and thus can't achieve upwards of 90% efficiency, that's not a reason to give up on hydronics, either!  Modern boilers are just as good as furnaces, if not better, at wringing every last BTU out of natural gas molecules: even your old oil burner can achieve 80% efficiency or more if tuned correctly, and modern condensing boilers easily match or beat what even the best furnaces are capable of when allowed to work to their full potential.

You get other advantages by sticking with a hydronic approach as well.  Indirect tank-type water heaters can provide you with a robust and efficient solution to your hot water needs, using heat from your boiler to heat hot water instead of requiring a second combustion appliance for the task; so called "reverse indirect" water heaters are particularly attractive due to their tankless-ish operating characteristics and ability to serve as a central "nexus" point for a hydronic system, providing the useful characteristic of hydraulic separation between various parts of the system.  This is very handy if you want to add fancier things to your system, like zoned heat, solar thermal collectors, or even what's called a desuperheater on your new heat pump (sometimes termed a "hot water generator") that uses extra heat from the heat pump circuit to heat hot water.

So, you want to use two different fuels...

While it is considered unusual to have a house that can switch heating fuels (between oil and gas or propane) on the fly, it's certainly not impossible, either, and the flexibility of hydronic heat makes this a much easier challenge than trying to introduce a forced-air furnace into the picture.  Once again, we go to the reverse-indirect, "nexus" setup described before, only this time, we use a fat ("low-loss" in the hydronic business) header or manifold at the supply-side ports, so that two boilers, namely your existing oil boiler and a new gas boiler (instead of the silliness that is using a gas furnace as backup for a heatpump), can both be connected to the reverse-indirect tank in a hydraulically separated fashion.  The rest of the system, then, can be as before, with baseboards, panel radiators, radiant floors, and/or air-handler coils as the various heating zones.

Given your preferences for fuel use during cold conditions (oil first, then gas), you'll want to run the boilers in a lead-lag setup, with the oil boiler leading and coming up to full fire, then the gas boiler kicking in for additional capacity if the oil boiler doesn't have enough grunt for the current conditions.  Furthermore, if the oil boiler breaks down, the lead-lag control will simply kick the gas boiler on in response to the lack of output from the oil boiler.  This gives you the redundancy you're looking for (and then some: it means that say, a disruption in oil supply won't leave you wishing for a hot shower), without having to deal with the issues involved with forced air, "dual fuel" heatpumps.

Nonetheless, if you insist on kneecapping your heatpump....

Nonetheless, if you are absolutely insistent on using a forced-air furnace with your heat pump, or somehow cannot find anyone who can implement hydronic system upgrades where you are at, it is possible to implement most of the control logic you desire using a fairly standard thermostat and some basic HVAC parts.

First, though, instead of using a rule-of-thumb for the changeover that's bound to be wrong for your system, we'll need to work out the actual Economic Balance Point for your setup, for each fuel.  You'll need the prices of the fuels you get (oil and gas) in $/therm (excluding fixed costs such as meter or truck-roll charges), as well as operating efficiencies (or AFUE ratings if you don't have operating efficiencies) for your equipment, the coefficient of performance for your heatpump at two specified temperatures, and the price of electricity in $/kWh in your area (including all variable charges, but excluding fixed charges).

Once you have the numbers in hand, you'll need to divide your fuel prices by your system efficiencies to get their operating costs, as well as multiplying the price of electricity by 29.3 to get a price per therm for electric heat at a COP of 1.  You then divide the electric heat cost per therm by the furnace or boiler operating price to get the economic crossover COP for that system.  Finally, you convert the specification COPs at the given temperatures into a line equation (remember this from algebra class?), and then solve that equation for the temperature that yields your crossover COP: that temperature is the economic crossover point for your setup, at least within a first order as the linear approximation of COP vs temperature fails to account for the "defrost knee" in the heat pump's performance curve.

Note that you will have two economic balance points in your situation, one for the gas furnace and one for the oil boiler.  Also, do not come into this calculation with preconceived notions of what heat pumps can and can't do: I ran an economic crossover calcuation for a recent Mitsubishi H2i mini-split in my area (which has relatively low utility rates all-around), and it came out below 0&deg;F! While the heat pump I used (a MSZ-FH18NA) is a stellar performer, with a COP >3 at 17&deg;F, I would not be surprised if 40&deg;F turns out to be rather higher than the economic crossover point for your system, especially on the oil side given that you have an older boiler there.

Also note that while the economic crossover point doesn't depend on load, your system's thermal crossover point does depend intimately on the heat load your house poses. Air sealing and insulation fixes are very good at cutting heating load, and are highly recommended in any case, but are especially beneficial in yours because they maximize your ability to use the heat pump as a sole source of heat.

Now that you're armed with this information, we can move onto setting up dual fuel.  You'll want to use a 4H/2C heat pump thermostat with integrated dual fuel support for this, not a separate dual fuel control board, by the way.  This way, we can run the oil system as the first stage of emergency heat and the gas system as the second stage.  You'll need to connect an outdoor temperature sensor to this thermostat, by the way, and also to wire it as a two transformer setup, with Rc/C derived from the furnace and Rh set up as switchable between being furnace-derived and being boiler-derived.  You'll also need a relay from W2 on the thermostat to the boiler C terminal that has its NO contact connected from R to W on the furnace board to isolate the boiler side from the furnace side.  Finally, you'll need DPDT switch connected with W1 from the thermostat on one COM terminal with the corresponding NC terminal going onwards to the boiler heat call and the corresponding NO terminal going to W2 on the thermostat, along with Rh from the thermostat going to the other COM terminal, R from the boiler going to the other NC terminal, and Rc from the thermostat going to the other NO terminal; flipping this switch disables the oil boiler as an emergency heat source.

Finally, when commissioning this control, you'll want to set the thermostat's compressor lockout (auxiliary heat changeover) temperature to the highest crossover point in your system; this is necessary for proper "dual fuel" behavior with the oil system cut out if your economic crossover points are below your thermal crossover point (a situation that's quite plausible).  If your economic crossover points are above your thermal crossover point, this is OK, but it means you'll want to recompute your economic crossover points each heating season, as it's very possible that your fuel price situation could change drastically on you.