Going to go more engineer than historian for a moment, but I might be least outline the situation for you. Much of it comes down to what you consider a failure.

Nuclear devices work by using a conventional explosion to slam plutonium together (usually imploding a hollow shell of it). When enough plutonium is crammed together, the neutrons being thrown around by occasionally decaying atoms get thick in the metaphorical air (that is, the space between atoms.)

On occasion, a neutron will slam into a neighboring atom and put it over the edge as well. When the material reaches such density that this chain reaction drives itself, we say it has reached critical mass. It explodes.

If the conventional explosion is imperfect, only a small part of the plutonium will get a chance to react before it blows itself to bits. You call that case a "fizzle"; pretty much every nuclear program has had a few. It can be a slightly - below - desired yield, or so pitiful it barely destroys its own support structure.

What I can't say would be procedure for handling a complete failure to detonate. The conventional detonator remains unsafe, so it would have to be treated as such; however, it's hard to say how the enormous danger radius would change the situation.

*I do need to note, this is just a rough idea. Independent neutron sources and the use of DT (deuterium-tritium) gas to literally dial in the desired yield have changed the game a bit since these monstrosities were conceived.

The ABLE shot of Operation BUSTER-JANGLE failed due to a wiring issue. This was Oct. 22, 1951. In this case even though it was a failure there was some yield and there was alpha-emitting debris spread around the test site. The overall description of Operation BUSTER-JANGLE can be found in Operation BUSTER-JANGLE 1951, Defense Nuclear Agency Report DNA-6023F, 1982-06-21. A description of the BUSTER shots can be found in Shots ABLE to EAST: The First Five Tests of the BUSTER-JANGLE Series, 22 October-5 November 1951, Defense Nuclear Agency Report DNA-6024F, 1982-06-22.

From the second report:

[...] the ABLE device partially misfired. It had a yield of less than 0.1 kiloton. The top of the cloud resulting from the detonation reached an altitude of 8,000 feet and moved southeast. (Ch. 2 introduction, p. 19)

So there was a bit of an explosion but not much. The altitude at the test site was around 4000 ft. above sea level.

[B]ecause ABLE's yield was less than 0.1 kiloton rather than the 0.25 kilotons expected, the AEC aborted many of the planned projects (§2.1.1, p. 20)

A helicopter survey of the shot area was conducted soon after the detonation. [...] The gamma radiation detected by the helicopter survey was negligible, and isointensity plots of the gamma radiation levels in the shot area were not prepared. However, alpha-emitting debris from the misfire of the device was scattered around ground zero. Within a 100-meter radius of ground zero, alpha activity was 300 to 400 counts per minute, with readings of more than 20,000 counts per minute on some pieces of debris in the shot area. Other alpha contamination was spread within an area about 700 meters east to south of ground zero. (§2.2, p. 23, my emphasis added)

Based on data obtained from the initial survey, the Test Manager decided to open the shot area for limited recovery operations about two hours after the detonation. (§2.2, p.24)

So they did some recovery to clean up the site a bit as opposed to leaving chunks of alpha-emitting debris around.

You also seem to be interested in shots that just didn't work at all. Perhaps the primary explosives detonated but didn't cause a nuclear reaction. I don't know offhand of any test shots where this happened however there were a number of documented accidents that have happened that caused the primary explosives to detonate. In these situations you have two potential outcomes: 1. No damage to the fissile elements. In this case it's safe to recover what's left of the device after a radiological survey has been done. 2. Damage to the fissile elements. Clean up procedures probably similar to what was done with the ABLE shot would be followed. I would read through this link for some examples.

Not related to your question but there have been "accidents" that caused a bigger explosion than intended. One of note is the CASTLE BRAVO test of an early hydrogen bomb that was twice the expected yield.

In general this sort of situation would be very unusual. If the firing signal did not do anything in the firing unit (which activates the conventional explosives in the bomb), then I don't think there's any chance of it just hanging around being any more or less dangerous than it was before the test. You'd power everything down and send someone out to check it out.

More likely is the conventional explosives would fire but no nuclear yield would be achieved. This is a "fizzle" and these happened occasionally. There were also fizzles that involved just being under yield, or those in which only the primary of a thermonuclear system fired but the secondary did not ignite. Neither of these would create situations with an "presumed active" weapon remaining.

There were plenty of ways in which an active nuclear bomb could be rendered quite dangerous in terms of switching, but for nuclear tests the conditions were much more controlled.