As Vietnhi Notes (this is, of course, becoming something of a go-to opening phrase for me), the name isn't as important as the work.
As you note, you "want people to know my skills are scalable". If this work does show that, then, yes, of course mention it. However, just because the client is large doesn't itself mean anything - large companies will have need of window washers just as much as small companies, the size differential between companies does not translate to one window washer being "better".
To that end, your phrase "During my time as a freelancer, I've managed projects for multi-million pound companies" is bad, in the sense that it doesn't express that you've done anything of note, but does read like bragging.
A better way would be:
Some multi-million-pound-company (date started-finished): managed the XYZ project, which successfully launched on time and on budget. The project contributed to 15% cost reduction in the ABC department, and 10% improved sales of widgets.
Some smaller company (date started-finished) : whatever you did here
and so on.
But this doesn't make sense to me - presumably you know how to write a CV, and linkedIn certainly leads you to input the name of the company, the date range, and what you did there.
Are you instead asking if, in that little intro space linkedIn provides, that you write "During my time as a freelancer, I've managed projects for multi-million pound companies?" It is still a waste of space as it sounds fluffy and braggy. A better way would be
"Specialist whatever-you-do freelance consultant. I've successfully delivered whatever-projects-you-do adjectives-on-time-on-budget-maybe for companies in the what-domain-is-this? ranging in size from small-company to -this-large-company"
Now it sounds like you do whatever it is you do well, and the large company is there to stress that you can work in all sorts of different environments. Much less braggy!