Can I use a LUT in a free video editor?
Yes. Video by Zaweba applies .cube LUTs with an intensity slider, has an input LUT slot for log footage and can export your own grade as a LUT.

You can use LUTs in a free video editor, and Video by Zaweba is one. It applies .cube LUTs with an intensity slider, has a separate input LUT slot that takes .cube or .3dl files before the grade, and understands the log curves of several camera makers. It's free software under the EULA, works offline and needs no account.
It can also go the other way and export a grade you've built as a LUT, so a look you make once can follow you to other projects.
What a LUT does
A LUT, or look-up table, is a file that maps every input colour to an output colour. Feed it a pixel and it tells you what that pixel should become. That makes it a neat way to package a colour transformation: a camera maker's conversion from log to normal colour, or a colourist's creative look, can be handed over as a small file and applied in any editor that reads the format.
Broadly there are two kinds. Technical LUTs convert footage from one colour space to another, typically from flat log footage to standard Rec.709. Creative LUTs apply a style. Video by Zaweba gives each its own place.
The two LUT slots
The input LUT slot sits before the grade. It's where a technical conversion belongs, so that exposure, contrast and the colour wheels then work on normal-looking footage. It accepts .cube and .3dl files.
The creative LUT is applied with its intensity slider, so you can use a strong look at a third of its strength rather than all or nothing. Most looks benefit from being turned down.
For log footage you may not need a technical LUT at all. Each clip can be given an input colour space: Sony, Panasonic, ARRI and Canon log curves, and HLG and PQ HDR. Wide gamut is converted to Rec.709 with filmic tone-mapping and an exposure control.
One limit to know about
The working pipeline in Video by Zaweba is 8-bit, so there's no HDR output. HDR and log footage come in and are converted to standard dynamic range. For web video, social posts, presentations and most personal and small business work that's what you'd deliver anyway, but a project that must be delivered as HDR needs a different route.
Using LUTs well
Apply the technical conversion first, then correct, then add the creative look. Check the result in View > Scopes, as a strong LUT can push highlights to the limit or swing skin away from the vectorscope's skin line. Twelve built-in looks are there too, also with intensity, if you don't have LUT files of your own.
To share a look, build it with the colour controls, curves and wheels, then export the grade as a 33-point .cube LUT. Apply that LUT across a series, give it to a collaborator, or load it in another editor that reads .cube files.
So a LUT workflow doesn't need a paid editor. The pieces are all in Video by Zaweba, from the conversion of log footage to a finished, reusable look.