In the case of the earliest tablets discovered at Vindolanda, I understand that the preservation was due to pockets of anaerobic (oxygen-free) conditions which had been created by layers of compacted clay between successive floor layers and the great depth of material above those layers (up to 6m in the case of the first discoveries) (this point is made on page 17 of The Roman Writing Tablets from Vindolanda by Alan K. Bowman). Preservation would also have been enhanced by the tannins produced by the organic remains and from the leather items that were also found in those layers.
However, you are mistaken about the oxidisation of metal artefacts. Even iron artefacts (like stylii for writing on wax tablets) in these layers were very well preserved. You can find details of the small finds recovered from these deep layers in the research report on small finds.
Having said that, tablets have also been recovered from later (and therefore shallower) layers at Vindolanda. In the case of these tablets, the mechanisms that created the (presumably) local anaerobic conditions that allowed preservation of the tablets, and the writing on them, do not appear to be fully understood and iron artefacts from these layers often do exhibit more oxidisation than those from the deeper layers.
(The recently discovered batch tablets that have been in the news over the last week were again recovered from anaerobic conditions in the deepest part of the site.)
The discovery and conservation of the tablets is discussed at some length in Vindolanda Research Reports, New Series, Volume II.
The report on the conservation of the original tablets is The conservation of the wooden writing tablets from Vindolanda Roman Fort, Northumberland by S.M. Blackshaw. It was published in Studies in Conservation, volume 19. I haven't been able to locate a (free) readable / downloadable version online though, and it is many years since I read it. Even when the ink is oxidised, and no longer visible to the naked eye, it can still be observed and photographed under infra-red light.
Vindolanda isn't the only site where tablets with ink writing have been recovered. The conservation of similar tablets from Carlisle back in 1981 is discussed in this paper. However, although it gives a fairly good description of the conservation techniques employed, it still doesn't address the underlying chemistry that causes the ink to oxidise and fade.
It's worth mentioning that the content of the tablets is available online on two sites:
Vindolanda Tablets Online which has details of tablets 118-573 from The Vindolanda Writing Tablets volume I and II, and some useful background resources. (This is the site you quoted from in the question)
Vindolanda Tablets Online II which has tablets 574-853 from The Vindolanda Writing Tablets volume III in addition to tablets 118-573 from volume I and II. This also has some useful indices and other resources.
However, neither site has any real detail on the conservation techniques used on the tablets, or about the chemical reactions that cause the ink on excavated tablets to fade.