Here, from the Bulletin of Atomic Scientists, a 1990 estimate of deployed nuclear forces in NATO by type and number (excluding as it states French forces): Page 1, Page 2. Ignoring the INF forces yet to be dismantled and to break down the various systems:

Since the early 1960s the nuclear artillery shells had been the 8 inch W33 warhead and the 155mm W48 warhead. These represented a significant improvement over earlier nuclear artillery rounds, which required larger caliber dedicated artillery pieces to use. However by the 1980s they were considered unsafe and unreliable. Replacement of them had been delayed, first in the 1970s when the next generation of the shells, W75/W74 had been cancelled in favor of pursuing enhanced radiation (ER, the infamous "neutron bomb") designs. These became the W79/W82. However these in turn then faced significant political opposition for European deployment, as well as opposition from the US Congress. This forced the development then of non-ER variants, in smaller numbers than planned. The W79 would be deployed to Europe while the W82 was ultimately cancelled. So the artillery shells in 1990 represented a mix of W33s and W79s for 8 inch guns, and still using W48s for 155mm guns.

The Lance missile was after the INF dismantlement were completed the only ground based missile left. Similar to the nuclear artillery, this missile had been an improvement over older systems but by the late 80s was showing its age. Its range was limited (especially compared to what was allowable under the INF treaty), its accuracy was poor limiting the effectiveness of its conventional variant, and it was deployed on a small number of launchers rendering it vulnerable to destruction. An agreement was made in 1983 for NATO to develop a Follow-On To Lance (FOTL) replacement missile. One possible replacement was a nuclear variant of the US Army's new ATACMS missile, a new conventional ballistic missile that was being developed for the MLRS rocket launcher. This allowed the MLRS, which was becoming a staple of several NATO armies, to serve as both a rocket launcher and a ballistic missile launcher. Given the capability of the missile a nuclear warhead option was proposed, but faced Congressional opposition in US, questions over its range as the FOTL requirements were expanded, and was never developed. Attempts to develop a separate new missile faced its own Congressional problems, but by now the entire concept of a FOTL was being heavily opposed in Europe, particularly in West Germany and ultimately all efforts at replacement were cancelled in 1990. The Lance therefore remained the only nuclear option until the end of the Cold War. The Lance also went through its own cycle of an ER warhead being developed and then undeveloped in the face of political pushback.

Finally there were the nuclear weapons deployed with NATO tactical aircraft. These represented a much longer range option for NATO. Even the short legged F-16 could outrange any of the remaining ground forces, while the F-111 had some capability to threaten targets all the way into the Soviet Union itself like the INF weapons could. A new aircraft coming into operation for NATO was the F-15E. This didn't have quite the range of the F-111, but it retained most of the air to air capability of the F-15C air superiority fighter, meaning it could defend itself much better as the survivability of low altitude strike missions became less and less certain against improved Soviet fighter forces. NATO tactical aircraft however still remained reliant on nuclear gravity bombs requiring a direct attack on a target. This weakness was supposed to be addressed with the development of the SRAM-T, a stand-off air to ground nuclear missile allowing an aircraft to strike a target while remaining several hundred kilometers away. It was going to be a variant of the SRAM II being developed for the B-1. This missile faced technical delays and was cancelled with the impending end of the Cold War which ended the SRAM-T.

Those are the nuclear forces deployed in Europe. However as the Bulletin states, there were forces deployed outside of Europe available for use in Europe. For the question of supplementing the lost INF forces, of interest are the sometimes overlooked assignment of submarine based missiles for NATO use, which had as much long range strike as needed, though with its old set of problems. As well there was the availability of sea based nuclear Tomahawks which were not covered under the terms of the INF treaty.

So to summarize, the tactical and theater nuclear weapons available to NATO after the INF treaty were largely the same as it had for a while. Short range ground based systems with longer range air and sea based options. Some new systems had been deployed, but many had faced considerable opposition. Part of this opposition was driven by the desires of further arms limitations, as after the INF success the next obvious target was an agreement of shorter range forces. However it went deeper than that. To go into that is how NATO expected expected to defend against a Soviet attack in the 1980s. That question deserves an in depth answer of its own, but unfortunately I don't really have the time and resources right now to answer it. But to cover it briefly, the idea that a Soviet attack could only be defeated with the use of tactical nuclear weapons was by the mid to late 1980s heavily contested. A new generation of NATO conventional weapons systems and new doctrines meant that NATO was seen by some as strong enough that it could defend a conventional Soviet invasion while remaining conventional. Again going into why various people were deeply unsatisfied with reliance on tactical nuclear weapons, the development processes of various new weapons systems (e.g. the US Army's so called "Big Five"), the fights over doctrine (Active Defense and AirLand and FOFA and all that jazz), etc. is beyond the scope of this answer. But to expand on an example I already gave, the US Congress pushed back on developing a nuclear ATACMS. Part of this was the normal reluctance to develop a new nuclear weapon and questions about the viability of the variant. But part of it was a desire to avoid any divertments of the funds for the development and deployment of the conventional ATACMS. It was seen as important new weapon that would improve the US Army's capability, and important to be fielded in large numbers, without the issue of a nuclear variant diluting that focus.

Partial list of sources: Tactical Nuclear Weapons and NATO, The Politics of NATO Short-Range Nuclear Modernization 1983-1990: The Follow-on-to-Lance Missle Decisions, The Bulletin of Atomic Scientists (Sept 1987, June 1990, Oct 1990 editions). Nuclear Weapons Databook, Swords of Armageddon