I have a feeling I'm going to regret this but I'll try and offer a simple explanation.....
Let's do the really simple thing first - ATC separates aircraft according to very specific rules. If two aircraft are in a situation where they should be separated under these rules, the controller's job is to make sure that the aircraft are at least separated by the specified separation.
WT separations are specified for pairs of aircraft and are based on the WT category (which is braodly based on the weight) of each of the aircraft. Depending on the situation, the minimum separation may be specified as a time or a distance. If a particular pair of aircraft are in a situation that has a WT separation specified, then ATC must ensure that the aircraft are at least as far apart as the specified separation.
Using the example that the OP gave, there is no WT separation required between two HEAVY aircraft taking off from the same runway under the ICAO (and UK) rules - so ATC doesn't have to worry about WT separation when issuing the take-off clearance. However, there are many variations - mostly minor - on the ICAO WT separation requirements applied around the world.
IFR separations are applied between aircraft that are flying under IFR. These separations may require that there is a certain minimum time between take-offs, particularly if the aircraft are following the same route.
A controller will look at each particular situation and work out whether either WT or IFR separation is required. If both are required, the controller will apply the more stringent (i.e. bigger) separation. As an example, radar separation (which is an IFR separation) in a particular area aproaching an airport may be 3NM but if there is a B747 being followed by a C150, the WT separation minimum is 6NM. The controller must then apply the more stringent WT separation 6NM.
At present WT separation either applies or it doesn't, depending on the mode of operation and pairs of aircraft, that is to say, the criteria for whether or not to apply WT separation (and, indeed, the separation criteria themselves) is not affected by other factors such as weather conditions. This will probably change in some regions before too long as research provides evidence that it is safe to do so.
There are differences in the way that WT separations are applied to aircraft that are flying visually or under VFR - and these differences vary from one State to another - but to consider them now is probably unwise when the basic rules appear to be at issue.
So, to sum up, controllers have a set of rules (separations) that apply in defined circumstances. If the circumstances fit two of these rules, then the controller will apply the more stringent of the two. If the rules don't apply to a particular situation the controller will happily sit there doing nothing and let the aircraft carry on doing what they want.
It's that simple - but if you start conflating different separation standards you will quickly confuse everybody!