While UC is unlikely to cover all your rent in most areas because the Local Housing Allowance (LHA) is invariably lower than average rents, each LHA area is pretty large. Where I live, the average rent for a 2-bed is £1,300, in the nearby city, it's even higher. But 7 miles down the road in a different town, you can get a 2-bed for £1,100, which is almost exactly what UC will pay. It's worth exploring exactly which places near you are cheaper. Things will get easier moneywise once you qualify for free childcare.
Are you on the council's housing register? And do they know you are sharing a room with your child? You should be a slightly higher priority because of that. If you can provide evidence that your housing situation is having an adverse impact on your MH, that might increase your priority rating too. And ask to speak to a housing adviser at the council, they may have a network of private landlords who charge lower than average rents that they can refer you to.
There are some housing associations that take applications directly, rather than only accepting people from the housing list. Google and see if there are any in your area. This organisation helps people find HA properties in many different parts of the country: Places For People .
The housing situation in this country is dreadful. I worked with people in housing need for nearly 18 years, but it got tougher and tougher to find affordable housing from approx 2012, and I made a sideways move into welfare rights in 2018. Nearly all my former colleagues have moved on because of burnout.
I don't understand why there isn't a national bloody outcry about it, tbh.