The NNFI is the original name for what is now most commonly called the TLI (Tucker-Lewis Index). You interpret it in the same was as you would the CFI.

CFI and TLI both penalize models for complexity, but in slightly different ways, and TLI does so more explictly. TLI adjusts the fit index based on the degrees of freedom, making it less likely to favor overly complex models. TLI is therefore more sensitive to complex models, because it is penalised more heavily, while CFI is less sensitive and more lenient in assessing model fit.

I have never heard of LT1 before in the context of CFA/SEM

I would never decide your model fits just based on one or two fit measures. First look at several, such as TLI/CFI, RMSEA, SRMR. These all provide different information. For example: RMSEA is sensitive to deviations of multivariate normality, so if you have >.90 CFI/TLI but high (> .10 RMSEA) there may be some structural non-normality in your residuals which might signal the need for multilevel analyses, etc.

When your model fits your data it also doesn't tell you whether it is the best model to fit your data. So make sure to check logical alternative models to compare your model fit to (with AIC for example in the case of non-nested models)

I would primarily look at the chi-square test of model fit. Virtually all of the other "approximate" or "close" fit indices have problems in terms of being less sensitive to model misspecification than the chi-square test.