The Israeli rabbinate is very particular in whose conversions they accept. There are Orthodox batei din (religious courts) outside of Israel that are not recognized as well. For that reason there are converts who will get converted by an Israeli Beis Din so that they do not run into problems later on.
Litvos, there are two issues that are under discussion here- first is whether the conversion is halachically accepted, i.e. a proper conversion by Jewish law. Conservative and reform Judaism has rejected much of Jewish law and do not follow the Torah; therefore their conversions would not be considered valid by Jewish law, which requires the convert to accept all of the Torah and its commandments.
The second factor that BaisYaakovLiberal brought up is a sociological acceptance once properly converted. She wanted you to be aware that the Orthodox community is often very insular, and often encourages people to be exactly the same. It’s a way to protect the community from outside influences and preserve our Heritage and moral way of life. But the downside is that people who don’t fit the mold can be excluded or otherwise feel that they are not accepted. A convert, or his children, may very well be affected by this. If you go through with your plan, you should speak with other converts and carefully chose the community that you want to live in.