Lechatchila (before the fact) it is not permitted to cook in the same oven and at the same time meat and milk dishes (see SA YD 108:1). A key issue is one of steam (zeiah, for details see here and there) which carries meat/milk flavor into other dishes. Indeed no opinion permits cooking open milk and meat dishes together in the same oven.
As such the vegetables in your question are considered basari and cannot be eaten together with milk. The simple solution however is to double-wrap one of the dishes (see here for more details) after which there is no more issue.
However, in the case of parve items who became basari through zeia, a majority of poskim (amongst which the Chida, Kaf HaChayim, Ben Ish Hai, Maharsham) allow eating halavi items afterwards without a waiting period with kinuach ve’hadacha (cleaning out one’s mouth and rinsing, e.g., through eating a bit of bread and drinking water or other drinks).
Therefore, to answer your question, these vegetables don't make you basari as long as you perform kinuach ve’hadacha after eating them.
Sources: after researching your question, I discussed it with R Binyamin Tabady who wrote up an answer in Hebrew (1, 2, 3) from which I took the above.