This page from the State Dept explains the K-3 visa process:
Given that you're in the UK, you should be able to file your petition with the London Consulate instead of having to go through the "lockbox" in the US. This may very well reduce your waiting time by a bit. More details here: https://uk.usembassy.gov/visas/famil...a-u-s-citizen/
and also here: https://uk.usembassy.gov/embassy-con...377.1557778845
Ultimately, you are the sponsor for your husband, though you can always use a co-sponsor like your mom.