Threads and patterns should match. But a pattern may not specify the thread, or just the number and not the manufacturer. Even if the thread is specified accurately: you might not have it in your stash nor available by your favorite suppliers. Now what?

Resize the pattern

A simplified rule of thumb is: look up lace in similar style you made earlier, use the same thread and resize the pattern until you get similar pin distances. Thus math could be avoided by trial and error.

You could avoid trial end error with a resizing table. But these tables usually rely on a number system. These are only accurate for threads of a single type of a single factory. For example with the table of Vibeke Ervø. If you know which thread was used for the pattern and what you want to use, you can read from this table how to resize your pattern.

Lists of compatible threads

Two ladies collected and measured the width of many threads suitable for lacemaking. The results are available in books:

Calculated conversion

Both ladies introduced their own definitions to express the width. By their mathematical definitions the values of BP and MWK could be converted as follows:
1000 / wraps per cm
= dD
1000 / dD
= wraps per cm
dD is a German shorthand for "average diameter"
This theory is shown below with the red line. The dots show the values found in practice. It appears that MWK winds tighter than BP. The blue line compensates for this.
The straight purple lines shows that 29 dD measured by MWK comes between 20-26 w/cm by BP. The straight green lines shows that 40 w/cm measured by BP comes between 15-19 dD by MWK.