Depends on how much (many?) data you have. If your sample is rather small, you'll probably need to settle for the classical test theory (CTT) assumptions. I've reviewed these in a few other answers to:
Basically, you'd treat your ordinal data as numeric, take the sum or average of individuals' responses to items intended to measure the same latent factor (which you refer to as a component), and use that as your score for the individual on that factor. Repeat for the other factors and proceed to calculate your correlations among the scores, if that's all you're really after. Schaufeli and Bakker (2003) seem to describe similar procedures for the 9-item UWES (but I've just skimmed it TBH), so this may be the best approach despite the assumptions it makes, because it would probably make your measurement model maximally comparable to that used in other studies with this measure.
If you have plenty of data, you might consider other methods of estimating factor scores like I've discussed on yet another answer to "Factor analysis of questionnaires composed of Likert items". chl's answer there is well worth considering too, since all sorts of response bias can affect Likert ratings. There's a surprising amount of correction you can perform with enough data.
Bias corrections aside, you could implement the factor analytic approach by calculating a polychoric correlation matrix of your 27 items. Feed that into a confirmatory factor analysis, and estimate correlations among the latent factors. Getting accurate estimates will require a lot of data, but these estimates won't involve as many assumptions as the CTT approach. They'll accommodate the ordinal nature of Likert data, estimate factor scores using only common variance among items, estimate how much unique variance each item has that doesn't belong to the intended factors, and estimate how much item covariance is explained by the model. That's a lot of extra info, if you have the data for it.
Schaufeli, W. B., & Bakker, A. B. (2003). Utrecht work engagement scale: Preliminary manual. Occupational Health Psychology Unit, Utrecht University, Utrecht.