Sounds like you'd just need a single factor to model the experiment, where if U corresponds to 0hours, and e.g. T has 2hours, 4hours 16hours, then you'd simply label the replicates with a single factor, with levels 0h, 2h, 4h, 16h etc. You obviously won't be able to infer anything about how the untreated samples evolve over time, so you won't be able to remove such effects from the analysis, so all your conclusions will have to be worded with this in mind. You say you've previously done the other, more complete design, which to me seems to offer a safer approach to meaningful biological hypotheses, so I guess it's cost issues that are driving the question?
The only option for an LRT with one factor is to compare ~time against ~1, which will look for genes that reject the null of being constant across all 'timepoints' (ie are the same across all the treated samples, and the same as the untreated).
DESeq2 is more than capable of answering comparisons between pairs of timepoints, including the 0h (untreated) vs 2h (treated), for example - for this you'd need the Wald test, rather than the LRT. Just remember to code the timepoint term as a factor, as if you code it as a numeric, the comparisons will look for linear changes of (transformed) expression across time, which mightn't be what you want.