From a biochemical point of view, heat detection is achieved by proteins at the surface of nerve cells. They basically just trigger a nerve signal above a given temperature. So they DO detect temperature and not a "heat flux". It may seem surprising that nerve cells react so quickly but the increase/decrease in temperature does not need to go all through the skin. It just need to be detected at the surface of the skin and a difference of 1°C is enough to start feeling a temperature change.
Interestingly, the temperature threshold can be changed by some well-known chemicals. For example, capsaicin (from hot peppers) will lower the temperature threshold for the heat-sensitive TRPV1 protein. This is what causes the burning sensation when eating spicy food. On the opposite, menthol (from mint) tricks the TRPM8 protein (and many others) into considering the temperature is lower than it actually is, which gives this sensation of cold in the mouth.
EDIT
The initial question took as an example the feeling we all know when we enter a cold room or touching objects. This edit is meant to address that.
It is true that, after some time, all the objects in a room will have the same temperature but our skin will not, just because our body produces heat and the air surrounding us is a poor conductor. If you take a thermometer in your fist, you should read roughly 27-29°C,(1) so let's consider that it is our skin temperature, for the sake of the demonstration. Also, I will consider that everything happens while temperature is 24°C max, and 22°C in the imaginary bathroom.
We can feel a big difference if we step on the bathroom tiles compared to what we feel if we take a wooden (or plastic) object in our hand. The tiles feel cold while the wooden object feels warm. A tile is usually a decent conductor of heat, so when we step on it, it rapidly cools down our foot sole and we feel it in less than a second. By contrast, a wooden (or plastic) object is an insulator. On contact with our skin, the exchange of heat is very slow, so the temperature of our skin will not change immediately and we interpret that as being a kind of neutral or warmish feeling.
(1) Of course it depends on many factors.
I am not quite sure what kind of examples you have in mind, the standard kinetic theory estimate of the thermal conductivity is
$$
\kappa = \frac{1}{3} n\bar{v} c_V l_{mfp}
$$
where $n$ is the density, $\bar{v}\sim T/m$ is the mean velocity of the molecules, $l_{mfp}$ is the mean free path, and $c_V$ is the specific heat. This says that keeping the mean free path fixed, thermal conductivity is proportional to the specific heat (and indeed this is the every day experience, a metal pot is a good conductor and has large specific heat, the wooden handle is the opposite).
Best Answer
Let me rephrase to:
Consider $1000kg$ of wood and $1000kg$ of aluminium, both at $320K$ (very warm). At the instant you place a finger on such large thermal masses, your perception of temperature comparison is dependent on heat conductivity of the materials, not their heat capacity (their masses are so large compared to your finger, their temperature is almost constant depsite losing heat to your finger). Using such large masses and (equal masses for that matter) is necessary since otherwise I can instantly answer yes to your question by giving you 100g of wood and 1g of gold (beaten to the same surface area of the wood) just taken from the freezer and you would perceive gold being closer to body temperature than the wood after a second. So lets define the question by specific heat capacity, and instantaneous perception of heat transfer.
To answer it though, there is in fact no metal which disobeys this relation due to the electron sea being the majority carrier of kinetic energy in the bulk metal. Their having large mean free paths and low masses allow them to attain very high velocities (which is a property of high temperature) and therefore are able to transfer energy quickly in the bulk material. In other words, if metals used anything heavier to transmit heat, like their nuclei, it would not only take much more heat to accelerate them to the same velocities the electrons could attain (resulting in higher heat capacity), but the rate at which that kinetic energy is transmitted across the material is accordingly slower (lower thermal conductivity). In fact the lattice of metal nuclei do in fact contribute to both properties via phonons not translational kinetic energy like in gases, but phonons are still greatly superseded by the effect from electrons. Therefore the inverse relation between thermal conductivity and heat capacity is valid for metals.
What you are looking for is a non conductor with both higher heat capacity and thermal conductivity than a conductor. For that I give you diamond (figuratively...I can't afford one), which has a specific heat capacity of $0.5 J/gK$, higher than that of any metal denser than vanadium (which is almost all of them), but has a thermal conductivity of $>900W/mK$, trumping silver's $421W/mK$ which is tops for all pure metals.
Indeed, $1kg$ of silver would feel much closer to body temperature than $1kg$ of diamond (that's alot of diamond!) despite diamond having a higher heat capacity.