I had a look through the conlangs in Wikipedia and listed which ones I could see morphologically marked TAME categories. There are sure to be some mistakes in here.
Tense: Atlantean, Dothraki, Esperanto, Glosa, Idiom Neutral, Interlingua, Kalaba-X, Kēlen, Kotava, Láadan, Langue nouvelle, Lingwa de planeta, Loglan, Mondial, Na'vi, Neo, Novial, Quenya, Sambahsa, Sindarin, Solresol, Sona, Syldavian, Universalglot, Uropi, Valyrian, Verdurian, Volapük, Wenedyk
Aspect: Kēlen, Esperanto (participles only), Ithkuil, Klingon, Na'vi, Quenya, Wenedyk?
Modality: Atlantean, Esperanto, Ithkuil, Kēlen?, Langue nouvelle, Solresol, Sona, Syldavian, Uropi, Valyrian, Wenedyk?
What's the verdict? Not only does this show that the conlang community has a strong Eurocentric bias, I think it shows it has a massive Anglocentric bias! Because it is English which is so strongly tense prominent. Most other current European languages also mark aspect or modality, but English doesn't even have a full aspect inflectional system, just the continuous -ing. Some European languages arguably don't even have tense, such as Koine Greek (for the last decade this has been a huge debate among Biblical scholars.) And evidentiality, which is present all around the world, including some European languages (some Slavic) was only present from what I could see in a single conlang! If Anglocentric bias were not a factor I would expect to see conlangs with a much broader distribution of inflected TAME categories, but as it is, 80+% have tense, maybe alongside aspect or modality.
In addition, many languages had muddled/fusional verb paradigms. You've probably seen them before: they include tense, aspect, the perfect, mood, utterance type (question or command), all in one table, each affix marking some combination of these semantic categories, but the combinations aren't very visible when they're listed one-dimensionally. Where are the agglutinative conlangs? Don't forget that European languages includes Turkish and several Uralic languages! (I found one conlang which is agglutinative in its verbal morphology: Einodo.)
Furthermore, often even when I judged a language to have aspect or modality it was because they had something called a "conditional" or a "subjunctive", rather than a core division between the prototypical categories: realis/irrealis, or perfective/imperfective. Now maybe some of those languages do actually have what should be analysed as the simple core categories, but they have been given alternate names by those describing them in Wikipedia. This reflects the Eurocentric and Anglocentric bias which is a big problem in descriptive linguistics. A helpful and short book about the importance of being aware of your biases in relation to TAM categories is The Prominence of Tense, Aspect and Mood by D.N.S. Bhat.