So confused with benefit system, can I claim and receive LCWRA right away?

https://www.gov.uk/government/publications/universal-credit-if-you-have-a-disability-or-health-condition-quick-guide/universal-credit-if-you-have-a-disability-or-health-condition

You would make a joint UC claim with your husband, and report your health condition with a Fit Note to start the Work Capability Assessment process.

You must continue submitting Fit Notes without gaps, (it is the date they are given to UC that counts, not the start date written on them) up until the decision.

Unfortunately your ADP has no impact on this process.

A WCA takes time to complete, if LCWRA is awarded then there is a waiting period of three full assessment periods* before LCWRA element is paid. No money is paid or subsequently owed for that period.

*Each UC payment is based on your circumstances for the previous calender month, known as the Assessment Period.

UC works by adding up entitlements, then making any deductions.

UC make a standard deduction of 0.55p for every pound earned.

If given LCWRA this deduction will start after you as a couple have earned monthly £411 (if claiming Housing Element) or £684 (if not).

It is possible with the wages no UC will be due, even if LCWRA was awarded and a work allowance added to your claim.

You might want to investigate New Style ESA which is a Contributions Based benefit. You can be awarded both (there would be one WCA the result applying to both benefits), the NS ESA is deducted in full from your UC award.

People do work & claim UC/ESA.

Some resources:

https://www.gov.uk/benefits-calculators

https://advicefinder.turn2us.org.uk/

https://www.gov.uk/guidance/new-style-employment-and-support-allowance