Yes, if you have had a backdated award of DLA, you can request that UC backdate all eligible elements to the date of award - you simply tell them you could not do the change of circumstances and inform them until you received the award letter. They will check with DLA/Child Benefit team when the award commenced to verify it once you inform them via the CoC section of your UC journal; then you can write on the journal, under Payments section, requesting it be referred to a DM for consideration of backdating - this may take some time especially if the award is a long one as they have to agree it then manually recalculate every UC statement from the one DLA was awarded in until present time.
If total payment divided by the time since applying works out to be around £300+ a month, that is MRC at current rates (£76:70/w x4) and would entitle you to claim carers element and lower rate disabled child element on UC. These would add around £365 a month to your UC entitlement.
If it’s MRC care awarded, you could also claim CA, but it is deducted in full from your UC as an overlapping benefit (the amount deducted is a bit more than you receive weekly from CA because you have to fit 52 payments into 12 UC assessment periods.
If you have 2, or more, children and your child’s disability means sharing a bedroom is difficult either for that child or their sibling/s, then you can ask UC to increase your housing entitlement so the child can have their own bedroom - but they will only pay it if you actually have the bedroom (so, if currently living in a 3 bedroom property, entitled to a 2 bed rate and have 2 children, they would increase your housing element to 3 bed rate; but, if in a 2 bed property, even if rent is above the amount allowed by UC for 2 bedrooms in your area, they won’t pay any more).
Regarding Mobility element, children under 3 can’t be awarded mobility element of DLA; between 3-r/s, they can only be awarded HRM and, only once they turn 5, can they be considered for LRM. There are routes for children with autism to be awarded HRM but the criteria are very strict and every single step must be evidenced by professionals and support within a variety of settings - have a look at the website of “Contact; the charity for families with disabled children” as they have a very good section on the eligibility criteria for children with autism to get HRM.