It is well known that the vowels were originally displayed as diacritics. In Rúmilian writing these have been attested, and it is shown how they modify the letter t. The sounds æ and weak ə were probably employed only in English.
Note that if the signs were placed on the left of the letter, they would be pronounced at, et etc.
Concerning the Modes using tengwar, the vowels were displayed with signs that are identical to all of the attested systems (with minor varieties). We can think that they were identical to the first ones that Fëanor made. The attested Quenya tehtar are shown here over the letter Tinco, and how they would be read in Quenya:
Sindarin had the ancient umlaut œ that later became e (as in Arnœdiad), but it is attested nowhere. Long vowels were placed on long carriers, but the signs for o, u and e could also be doubled. In Quenya, the sign of a could be simplified as a circumflex or even ommited, as it was a very frequent sound. Tolkien says in some modes, the signs for e and i swapped values, which is seen in the Ring Inscription, but it's not clear if this swapping occurred in Elvish modes. This was also the case with the signs for o and u in the Ring Inscription, because in Orcish, the sound o was rare.
We don't know how dipthongs were written in Sarati and Valinorean Tengwar, but in later Quenya they are written in a peculiar way: The tehta is placed over the following element. Some of them are attested in Namárie, the others formed by analogy. The six diphthongs of Quenya are:
And of Sindarin (tehta mode):
And both full modes:
Au and oe have been attested nowhere and are guessed. Oe shouldn't be confused with the ancient umlaut œ. Au is written -aw finally (as in baw, lhaw) by Tolkien. Final -w in other combinations like têw, aew, lhiw is a semivowel and doesn't form a diphthong.