In a scenario where the Soviets didn't manage to stop corium from reaching the water reservoirs underneath the plant, causing a detonation equivalent to a nuclear explosion, what would life in the continent look like?
There would not have been a nuclear detonation as the core materials mixed with debris would not have been fissile. And even had they been it would have been more of a damp squib than a true nuclear detonation.
That said even a conventional detonation caused by steam and chemical reactions would have been disastrous as a much larger part of the radioactive contents of the reactor would have been spread around the environment, probably at least an order of magnitude greater, but it’s very hard to be certain as it would depend on the exact circumstances.
The net effect would have been a much larger exclusion zone, greater loss of life due to radioactive contamination and even more money required for the clean-up.
But bad as it would have been life would have gone on across Europe.
causing a detonation equivalent to a nuclear explosion
Such a detonation could not happen.
At most, when the lava1 reached a reservoir of water, it would turn it into steam very quickly. Depending on how stiff the structure was, and how long that very high-pressure steam could have been contained, you could have a Boiling Liquid Expanding Vapor Explosion (BLEVE). The most likely case will be that the steam will push a large chunk of the molten concrete and fuel-rod mixture into the air, which would be little different from what had already been happening.
The concussion from the BLEVE might kill people more quickly, but if anyone was close enough to be killed, they're probably among the many who had already received lethal doses of radiation, and this would be a merciful death compared to what awaited them.
Molten fission fuel can not have a "nuclear explosion" in the sense of an explosion from a nuclear bomb. In order to go critical -- to have an ongoing chain reaction -- the fuel needs to have a certain density. As the fuel becomes more dense, it heats up, which pushes the fuel apart and halts the chain reaction. This can go on for quite some time, and it did indeed go on for several days after the initial explosion, but it is a slow burn rather than the milliseconds-long reaction that creates kiloton and megaton explosions from fission nuclear weapons.
The real danger was, if the lava did break through to under the foundation of the power plant, the groundwater would be contaminated far more than it already was, much more than the runoff from surface contamination and fallout. The groundwater flows into the Dnieper River that supplies fresh water to Kiev, and the USSR had plans for at least a partial evacuation of that major city in case the foundation was breached.
1Pedantic note: The term lava is a correct term for the mixture of molten rock, cement, metal, and fuel rods. Another term that can be used is corium, which is a lava created by the core of a nuclear reactor melting down. I use the more generic term because the radioactivity has no significant effect on the behavior of a BLEVE.
It would not have been much different than what happened IRL. You see, while hot-in-both-senses uranium contacting cold water will create an explosion, it will be far from catastrophic. This is because those water reservoirs are (a) underneath the plant, and (b) the uranium would come down at a trickle. The net result of these two factors is that, instead of a "Oh hey, 20 tons of ultra-hot uranium just dumped into the cisterns and by the way we are all dead from the resulting explosion", you just have a "Oh hey, there was a minor explosion in the cisterns just now, but it is completely inconsequential compared to the NUCLEAR FALLOUT ARMAGEDDON WE ARE ALREADY EXPERIENCING!!!
The net effect of all this is that the exclusion zone (the area that you can't live in) would have been a bit larger. Honestly, considering all the FUBARing that was happening anyway, I doubt that anyone would have noticed the reservoir contamination until after the fact.
Bad, but not apocalyptic.
With ionizing radiation, any amount increases cancer risk, etc. There is no safe dosage. On the other hand, spread over Europe or the entire Earth, fallout will be widely dispersed.
According to this, Chernobyl released 5% of the nuclear material to the environment. According to this, Chernobyl equals 1% of the worldwide nuclear weapon tests. Of course not all isotopes are made equal, but a total release of the material in Chernobyl would be on the same order of magnitude as nuclear tests.
Those tests were a bad idea. In all likelihood, they caused plenty of deaths by cancer. More in some places than in others, like the well-known John Wayne example. They didn't make Earth uninhabitable.