It is logical to put resources where they are most needed. However, think about if a decision needs to be made, whether a temporary decision could be made or whether a decision can be delayed until a later date. Unless there is a directive or temporary legislation from Government that exempts the use of meeting statutory duties at this time and allows decisions to be made that would otherwise have required consultation, we must assume normal processes should be undertaken. You might consider whether you can suspend a decision and consultation until later. If that is possible and will not have negative impacts on people, that would be the logical course. If decisions need making because to not do so would have negative effects you might seek legal advice as to whether under these circumstances you can make a temporary decision and then, when emergencies are over, consult upon it. If you cannot make an emergency decision, whether because it might not be lawful or because it would be hard to justify, then you need to undertake consultation in the best possible manner under the circumstances and within the resources you have.
If something is not statutory, you must consider whether it would have negative impacts on people or affect a lot of people. If it would, then people have a legitimate expectation to be consulted. If you have usually consulted on such non-statutory matters, then people might cite a legitimate expectation to be consulted. So, if a legitimate expectation exists then you should treat the scenario as you would a statutory one (written above). If not, then consider moving ahead without consultation. But you might consider some engagement and do not forget that this will not absolve you of the need to do equality analysis if it relates to new developments or changes to the services you provide, whether statutory or non-statutory.