Better not — you might lose them. Make copies of your passport and BRP and carry them just in case if you're worried they might be needed. (They probably won't be.)
A lost BRP is a huge headache since it is your visa. Without a BRP, it will be very difficult to re-enter the country, and replacement can take quite a long time. Don't let this happen.
You can also carry a donor card with your code and surname if you've signed up for the organ donation program (don't be stingy, sign up).
The only exception is buying alcohol in shops or bars. If you look younger than 25, they may ask for ID.
The hardest part is convincing the airline to let you board. Border officers will let you through without a BRP.
If you can't convince them, you can apply at the embassy for a temporary entry permit.
Police registration at OVRO is currently suspended.
You probably need to register with the police after arrival. You can check on the police website.
Initial registration is done at OVRO. For registration, you either book online or go to OVRO and take a ticket (and then stop by Borough Market, where there's lots of delicious food — you didn't come for nothing). Or you can arrive early, and if there aren't many people, you might get through without an appointment.
During quarantine, registration appointments are suspended. But you must register when it resumes. Even if that's in a year. The main thing is to sign up on the website. Lack of available slots at OVRO is a valid reason for not having registered within the prescribed timeframe.
For registration, bring your passport, BRP, and a completed form with a photo. Check the list on the website. You'll need to pay a small fee on-site for registration.
After registration, OVRO will give you an A4 registration sheet. So it's better to bring a folder so you don't crumple or lose this sheet on the way back.
After changing residence or workplace (this applies to family members listed as dependents too), you must register again, but locally. No need to go to OVRO.
Generally, the police are more inclined to warn than fine, and there have been cases where people re-registered a year after moving. But it's definitely not worth risking it. And for obtaining ILR (indefinite leave to remain), you need to be an exemplary resident.
At photo shops, you can get photos taken and print documents from a USB drive, but prices are steep. Documents can be printed for a small fee at libraries.
You probably won't need this for the move itself, but later you may need to make copies and have them certified.
If the document is being certified for another country, you'll likely need to go to the embassy — there's usually a notary there.
For the UK, you can get a copy certified at a post office.
When you set up your UKVI account, you registered your previous BRP and this is no longer valid.
You can change this by logging on to: https://www.gov.uk/update-uk-visas-immigration-account-details
Once you have updated your sign-in details to your new BRP, you need to return your old BRP to the Home Office.
The BRP card should be cut into quarters and posted (in a windowless plain envelope) to the following address:
BRP Returns, P.O. Box 195, Bristol, BS20 1BT. You may have to pay a penalty of up to £1,000 if you fail to return an invalid BRP.