I have got OpenWith.exe working in WinPE 10, build 10.0.586.122, both 32-bit and 64-bit.
Only thing I haven't got working is the icons displayed next to the 'Open with' programs - I will update when I figure this out. The program itself works fine.
Registry dependencies
The below CLSID keys and their subkeys are accessed by a functional OpenWith.exe. Note that not all of these may be necessary - these are just the 'hits' I identified during analysis. I have added the necessary keys to a .reg file here (configured for HKLM\software) - http://www.mediafire.com/download/qh4i5tp5wd5ax55/OpenWith.reg
NOTE: I had to deploy the above from startnet.cmd. For some reason I couldn't add it during image build (some kind of odd permissions bug?)
All the other keys were already in my base image.
{00020424-0000-0000-C000-000000000046}
{0DEC7D0C-BD63-4759-AA64-ADEFD70DCF9A}
{1F486A52-3CB1-48FD-8F50-B8DC300D9F9D}
{20D04FE0-3AEA-1069-A2D8-08002B30309D}
{2155FEE3-2419-4373-B102-6843707EB41F}
{228826AF-02E1-4226-A9E0-99A855E455A6}
{4DB26476-6787-4046-B836-E8412A9E8A27}
{54E211B6-3650-4F75-8334-FA359598E1C5}
{603D3800-BD81-11D0-A3A5-00C04FD706EC}
{66742402-F9B9-11D1-A202-0000F81FEDEE}
{6E29FABF-9977-42D1-8D0E-CA7E61AD87E6}
{76765B11-3F95-4AF2-AC9D-EA55D8994F1A}
{83D6F579-4A45-439F-994E-7EC23C46B13E}
{83E94DBF-7F97-46B0-A6F0-360FE982BF83}
{94B23D4D-1040-4C4B-9081-85D8D6FA36C4}
{9AC9FBE1-E0A2-4AD6-B4EE-E212013EA917}
{A2C25004-96F4-4D49-B38E-6A52BE7E2F51}
{B196B286-BAB4-101A-B69C-00AA00341D07}
{B77B1CBF-E827-44A9-A33A-6CCFEEAA142A}
{C2F03A33-21F5-47FA-B4BB-156362A2F239}
{C90250F3-4D7D-4991-9B69-A5C5BC1C2AE6}
{D25D8842-8884-4A4A-B321-091314379BDD}
{E44E9428-BDBC-4987-A099-40DC8FD255E7}
{F324E4F9-8496-40B2-A1FF-9617C1C9AFFE}
DLL dependencies
I had to add the following:
\system32\OpenWith.exe and its MUI
\system32\twinapi.appcore.dll and its MUI
\system32\windows.ui.immersive.dll and its MUI
A number of other modules are loaded, but all of these were present in my base image.