Wow, I always thought Unitarian Universalist
was a Christian denomination, just like the Quakers?
My family went to a Methodist church when we were younger but are varying degrees of lapsed. My brother is a hard-core INTP atheist who actively looks down on Christians and doesn't believe anyone
reallybelieves in God. I am one of those hippie hybrids of CCM lovin' spiritualists (?)
Our responses when asked about church/god are below:
Me/ENFP female: For strangers simply reply, "I don't go to church?" Why not? "
I just don't go to church/Hmm, not sure" followed be tense smile or distracted look --> the other person gets the hint 75% of the time. They might add a final, "God wants you to go to church, etc." and I just nod.
If they really push I often lie to avoid the drama. I told this one woman who was way persistent about my parents, "
They're Buddhist" and she looked me straight in the face and said, "We have to convert them". Hahahaha. She also told me that even though I look like a college student I'm an old lady and I need to hurry up and get married and have some kids.

She was a total stranger. My people are funny sometimes -- but for the record, the Buddhist ones are
waaaaaay more laid back.
With people I know? When I get invitations to go to church at work I just say "
Wow thank you for asking me". I mean, it is nice to get the invitation I'm just most likely not going to follow them up on it.
With friends or people whom I would like to befriend, that's when I feel invested/motivated to talk about it.
I view invitations to church not as opportunities to talk about religion so much as old-fashioned social and friendship overtures. It's flattering to be invited to someone's place of worship. And you can still be friends (hopefully) with these individuals w/o attending their church or sharing the
exact same faith.
For you and your situation? I think the easiest (though not necessarily smoothest) approach is to be blunt like my brother/mom, but coupled with my more friendly vibe. Just say, "
Thank you but I'm Unitarian Universalist" or just say, "
Thank you but I have my own church that I attend regularly". That might open up conversation about your faith if you want it.
Also, is it against your faith to attend other services? Sometimes going to someone's place of faith is literally a "good faith" gesture, it's friendly and social, even if you aren't of the same faith (I dunno, I attended a Jewish Shabbat service once and I've visited Buddhist temples and no one seemed to care)
And just becuase I bothered to write them -- more blunt/funny responses:
INTP 20ish male Christians
piss him off: One woman at work kept trying to convert him/get him to go to church. He interrupted her, said bluntly,
"I don't believe in God" and walked away. The woman literally choked and sputtered "Wha? Wha? Wha???" Hahahhaah. So wrong. So (young) INTP (male).
Assertive IXNX Mom: This woman at the hair salon was pressing my mom about church and god and my mom told her, "
My relationship with god is fine so I don't have to go to church and I'm done talking about this."
BTW, probably not applicable to you but funny to me and my friend -- some Jehovah's Witnesses or Mormons (I forget...) went to her mom's house who is Buddhist and she sat them in the living room where the huge Buddhist altar is and she lectured them on the story of Buddha until they excused themselves and left. LOLZ.