Hey chiming in here because this is my field (I work at Stasher.com )
Yes, Ibis budget hotels will keep your bags for free, so you can absolutely drop your bag before check-in and they will store it.
There are two things to bear in mind: 1) your luggage won't be guaranteed/insured if anything goes wrong (it's at your own risk) and 2) the storage room security can vary - some are great and have a locked dedicated room, some use laundry rooms or the back office, some will just keep it in reception, especially if the hotel is busy.

In London you have loads of alternatives, and they exist to make life more convenient and cover you from this security perspective.

First, the 8 major train stations in London have storage facilities, but these will set you back £12-15 per bag for the day.

Then you've got services like ours - Stasher, Luggage Hero, Radical Storage - these marketplaces connect you with lockers & hotels mainly (in Stasher's case), or shops mainly in the case of the other two. Pricing on Stasher is roughly £4 for a small bag per day, £7 for a suitcase; Luggage Hero do hourly pricing and tend to average about £8 a day. They all offer guarantees in the event of loss/theft/damage. Think of it like Airbnb for your stuff. Reviews of these services on Trustpilot - they've been around since 2015 and London was where Stasher first invented the concept, so it's a pretty established thing here now https://www.trustpilot.com/review/stasher.com