Optimal procedures may differ depending on how many ordinal variables you have,
how widely the number of categories differs from variable to variable, and whether
all variables are of equal importance.
The following tentative suggestion may work best for a few variables of equal importance and numbers
of categories between 3 and 7. That is, labels $1,2,3$ (fewest) and $1,2,\dots,7$ (most). I offer this suggestion with hope it may give you ideas for an improved
version or a better method.
Roughly speaking, the idea is to re-scale all variables so to have lowest label $1$ and highest $7,$ treating variables as if they have interval scales, so that label
"equally spaced" to respondents. And then "averaging" or taking the median, depending
on how comfortable you are with the 'equal spacing' assumption.
In particular, for the specific example in your Question:
- Item 1 scored 2 on a 3-point scale becomes $(7/3)2 = 4.67.$
- Item 2 scored 4 on a 5-point scale becomes $(7/5)4 = 5.6.$
- Composite score is average $5.13$, perhaps rounded down to $5$ on a 7-point scale.
This may be about right because the composite of a middle score and a next to
highest score is a score slightly above the middle.
Most important, with whatever method you decide to use, look at a few dozen composite
scores to see if results seem sensible. [If some scores in the composite are more important than others, you may want to consider weighted averages. Perhaps look most carefully at cases where component scores vary between low and high.]
Note: One could argue that my suggested method is more fair if scales
are treated as if they start at 0 (so a 7-point scale runs from 0 to 6, etc.). Then treating scores as if they were ratio-numerical, the computations in the example above would be $(6/2)1 = 3, (6/4)3 = 4.5,$ and composite $3.75,$ which might round up to $5$ on the usual 7-point scale. My (very limited) personal experience is that, when the versions differ, the one described in the main part of my answer works better.