Apparently nylon Lyre strings are not readily available but I was able to locate one on-line seller offering the equivalent of nylon Lyre strings. From the site:
These modern Fluorocarbon strings are a better alternative to plain nylon and are very popular with harp and ukulele as they produce a clear and strong sound, On lyres this is very effective on Plywood or thicker topped lyres that are otherwise dull with guitar nylon or too quiet with gut. Once settled in they are very stable in tuning, impervious to moisture these strings are usual first choice on my Standard models Lyres
Here is a link to the site.
Nylon Lyre Strings For Sale
Another option is to extrapolate from guitar string gauges as steel strings compare to nylon strings. For example I found this on a web site that sell's Lyres.
Looking at a set of standard gauge classical nylon guitar strings I find this for listed string gauges:
Comparing that to a similar set of plain steel for electric guitar (so we get a plain steel G string) I find this:
The common denominator between Lyre strings are the .12 the .16 and .20 which correspond to the 1st, 2nd and 3rd string from a plain steel set of guitar strings. Extrapolating from there to the corresponding nylon guitar strings we get:
1st string - Steel .012 ... Nylon .028 = a 2.3333 multiplier
2nd string - Steel .016 ... Nylon .032 = a 2.0 multiplier
3rd string - Steel .020 ... Nylon .040 = a 2.0 multiplier
So I think we can conclude that assuming the relative tension of nylon vs. plain unwound steel guitar strings to allow for equivalent tuning - is a 2 to 1 ratio in terms of gauge - you could take the gauge of the steel strings in a set of Lyre strings and simply double the diameter for a corresponding nylon string.
If it works with guitar strings I see no reason why it would not work the same with Lyre strings. It's really just a matter of which gauge nylon string tunes up equivalent to which gauge steel string. The overall tension will be lower on nylon but the relative tuning properties nylon vs. steel strings should be the same regardless of what instrument the strings are on.
Good luck and enjoy playing your Lyre.