A linear change across two timepoints is actually exactly the same as a difference in means between the two timepoints, so there is no need to change that language. You still have found a "significant difference" and it's perfectly fine to describe it as that. However, all that significance means is that your dataset would be fairly unusual if there were truly no difference between time V1 and V2. Which actually isn't that informative! So, for sure, it's a good idea to report the B and its SE too. But, for the reason that they tell us much more than a $p$-value; not for any reason that is peculiar to the mixed model.
(For example: your Wilcoxon provides you with a pseudomedian difference and its confidence interval too, which are prodding at very similar things to the B and SE of the mixed model. Again, they are much more informative than a $p$-value).
Regarding the controls, yes I think you describe it well! You are seeing how each of the three things (Time, sex, age) affect T1. (But, in case I've read your sentence wrong: you are not asking if time-differences are impacted by sex and age; that would require you to add an interaction term to your model).
In most cases, adding those extra controls in the model also would help it to tidy up the T1 variable, so that once sex and age effects have been removed, the model gets a clearer view of the differences across time. Because you have paired samples, I think I'm right in saying, age and sex don't add anything in this sense and if you removed them, the $p$-value on Time should stay about the same. Because you already have a random effect that controls for the average of each individual, and the effect of their age and sex is already captured in that. (This is assuming that ID captures participants and the ages/sexes are the ages/sexes of these same participants).
So Why Do the $p$-Values Differ?
If the controls are indeed not controlling for anything, then the time-component of your mixed model is now roughly equivalent to a paired $t$-test (I suspect a paired $t$-test should give the same p-value as the mixed model). And the differences in $p$-value (compared with Wilcoxon) is really down to the parametric vs non-parametric thing.
Both tests try to answer fairly similar questions in (in my opinion) quite similar but, nonetheless, different ways. That one is significant and the other not is no contradiction, because a non-significant result is not the opposite of a significant one -- it is just one that didn't quite reach a threshold. It is a reflection of the fact that a slightly different angle can colour things slightly differently. And even more, it might be a reflection of the fact that a hard line at $p=.05$ isn't really meaningful. If you compared the two models in terms of B and its CI for the mixed model (you can get that by running
intervals on your mixed model) and the pseudomedian difference and its CI (which you can get by setting
wilcox.test) it's quite possible that the results won't differ that much at all...
(The best approach would, of course, be to use just one approach in the first place, so that we are not tempted to choose the $p$-values that suit us -- a.k.a. $p$-hacking).
By the way, I'm not super familiar with the
nlme package, but I think you want the
lme function in the
nlme library. (You are using the
nlme function at present, which is for non-linear models, I think.) Also, I think your random factor needs to be something more like
random=~1|ID, but again, not super-familiar, so that might be wrong.