I will give an overview based on the analysis of Chapter 6 of “Monetary Economics: An Integrated Approach to Credit, Money Income, Production and Wealth” by Wynne Godley and Marc Lavoie. This is a Post-Keynesian textbook; neoclassical economists might debate the exact model structure, but their models probably give qualitatively similar results.
There’s a lot of issues buried in your question. One immediate problem is that we should not really look at just the government deficit: what the government is doing also matters. For example, the government could decrease spending in aggregate, yet greatly increase its imports (e.g. defense spending) and so the trade deficit could get worse. We will assume that something like that is not happening.
Arguing that government spending “will not impact savings” (as the question is now stated) is incorrect. Savings desires might well be unchanged, but changing government spending changes incomes, and so actual saving will change. Furthermore, cutting spending presumably has some multiplier effect (the precise level of the multiplier is hotly debated), and this will affect investment plans.
In the model in “Monetary Economics,” cutting government spending reduces the trade deficit. Since it is a multi-equation system that is solved simultaneously, one needs to be careful about verbal descriptions of mechanisms. However, I would describe the mechanism as follows: reduced government spending reduces national GDP, and this implies less imports (since imports are a fixed propensity to import out of all spending). As a result, the trade deficit falls, but it is not 1:1 with the reduction in spending - as your question phrasing (at the time of writing this answer) suggests.
One could debate the model structure, but I cannot think of many model structures that would suggest that the trade deficit would widen in response to government spending cuts. The implication is that the real debate is about the coefficient that relates the size of the cut in government spending to the trade deficit reduction. In almost any model, that coefficient would be parameter dependent, and so one would need to look at observed data to come up with a prediction for the real world.