By Unicode principles, a combining mark must be appear (in data stream) after the character it is to be attached to, so the correct notation would be ω̑.
Browsers often have difficulties with placing combining marks properly, though modern browsers tend to do it mostly well. Here the issue seems to be with the font. Fonts often have inadequate implementations of combining marks, especially as regards to metric information. You can see this by testing the string, ω̑ in different programs (e.g. MS Word), using different fonts. E.g., in Cambria, the rendering looks OK.
Before considering the choice of a font, or a font-family list, consider whether you really want to have an omega with inverted breve. Such a character does not appear in any form of Greek writing or mathematical notations that I know of. You might actually mean omega with perispomeni, ῶ, which is a normal character in polytonic Greek; it would best be written as a precomposed character, U+1FF6 GREEK SMALL LETTER OMEGA WITH PERISPOMENI, using the character reference ῶ if needed. You would naturally need to check that all the fonts in your font-family list contain it, but this is a manageable problem.
Addition: According to the Unicode Standard, clause 7.2, “U+0342 combining greek perispomeni may appear as a circumflex , an inverted breve ̑ , a tilde ̃ , or occasionally a macron ̄ . Because of this variation in form, the perispomeni was encoded distinctly from U+0303 combining tilde.” So it seems that at the character level, the correct diacritic to be used is the perispomeni. The fact that its shape is tilde-like in Times New Roman, as well as in most other fonts, is comparable to other glyphic variation.