It depends on each LA's way of doing things. They may well have different general policies for shorter 'introductory' profiles, and the full Child Permanence Reports.
Some will be happy to show you more than one short profile at once, wheras some will go for one at a time and ask you to decide whether you would like to see that childs CPR before showing another. Depending on how likely you are to find an in-house (within your LA) match within a few months, you could of course also subscribe to the waiting child magazines or use AdoptionLink which will give you access to a lot of short profiles at once
Most agencies will have a 'one CPR at a time' policy but not all of them - I was allowed to be looking at two CPR's at the same time way back when I was searching for a match.
There are a few 'adoption party' events in the country, which is an entirely different way of doing things, where you meet the (harder to place) children before requesting CPR's, but this is uncommon. Also uncommonly, a few LA's will allow prospective adopters to see the child they're linked with before the official matching panel. IME, that tends to happen when the child is harder to place or something has gone wrong before (eg. previous adopters backed out during introductions) so they want the new set of adopters to be as certain as possible.
Normally though, you don't meet your child until the very very end, so I'd assume that's what will happen for you. The (very generalised here) sequence of events - You see the CPR and possibly photos or a short video, want to take it further, meet the childs SW, the SW likes you, you might ask some further questions, the SW team have a meeting and agree on you. Now you're linked. Now you find out even more about the child, and still being happy, you meet people, eg. medical advisers, foster carer, teacher maybe. You aren't officially matched until the matching panel agree and the decision has been ratified by the decision maker, so your matching panel will be booked in. Matching panel happens, and they agree with the match. Now the only thing left is to finish planning introductions. They can begin very soon after MP sometimes, within days, or within a couple of weeks maximum usually. Just before you meet your child, you'll begin your leave from work, and the intros will be very full on. You meet your child for the first time for a shorter while (a couple of hours maybe) and then the next day, and nearly every day after that until they move in, which could be within a week, usually within 2 weeks depending on the childs age and needs.
I'd say the matching process from first seeing the short profile to intros normally takes a couple of months - 2-3 months maybe? That's a very rough number, it varies depending on circumstances, but it does take weeks.
So basically, you committ to adopting your child before meeting them, based on all the information you've been given.