they warned me that the edges might not be perfect
This depends on how they plan on enlarging the print for cutting. If they keep the ratio to fit the height of 4", then .6667" will be cut at the max width (.33" from the left and .33" from the right). This may be a problem depending on how the design was made. This will most likely be an issue if your design has a border that is a different color than the background.
Can an oval designed for a 3 x 5 space fit exactly in a 4 x 6 oval when enlarged for printing?
No, the ratio is not the same. Enlarging and keeping the ratio of a 3x5 to 6" wide gives a height of 3.6". If they enlarge the print to be 4x6 then the graphic will be distorted vertically by .4".
but i am worried about the text being skewed
If they distort the ratio to fit the graphic into the 4x6 design space, then that is a personal preference if that is a problem.
The red outline & text shows the graphic stretched to fit 4x6. The black outline and text shows the graphic enlarged to 6" wide but keeps the aspect ratio and has a height of 3.6".
Should i trust the printer?
You can ask how the graphic will be effected by the enlargement and maybe they will give you a proof/electronic proof but we don't know the printer, what the design looks like or how they're planning to print the design so I cannot give advise for this.
In the end, if you have some time and this is an expensive project, I would wait for the designer to fix the graphic for the new dimensions.