@Tonee77. Some random thoughts to try to answer what I think your questions are around, and hopefully relevant to the OP (I think for more specifics you might want to start a seperate thread)
All (well almost) UK adoptions are through an agency. Voluntary Agencies are charities that only recruit and match adopters. Local Authority Agencies do the same, but mostly (by no means always) with babies taken into care by their Children's Social Care section of the same LA Social Services department.
Children are matched through what is effectively a moneyless market. Young, non-poor, educated, healthy adopters with a good support network and experience of children have the most value. Healthy 'easy to place' babies are the most in demand adoptees.
(Usual caveat: by describing the process in these cold terms I in no way suggest the system is bad; but it can be unforgiving)
If you want a young, healthy baby, it is likely that an LA will give you a better chance of being matched quickly. Fewer 'easy to match' babies are matched though VAs recently. There is HUGE variation between agencies; both in the quality of training and support and in the process of matching. The wrong agency can slow the process and make it harder, but you get there in the end. The 'supply' of children, particularly babies, varies hugely across the country. I have no idea about London, but I would contact several agencies to get an idea.
Broadly speaking, if you mostly match the description I gave above, you should be able to match with a baby within 12 months of approval at the outside. If (for example) you were in an area with few babies, you might be better with a VA that would search nationally. Most babies these days are placed through Foster To Adopt- worth researching this if you go further.
Broadly speaking, if you are nothing like the description above (i.e. older, less educated, poor, single parent, health issues, or other barriers) you are not going to end up with a perfectly healthy baby. Obviously there are many, many exceptions to these general rules. Furthermore, getting an 'easy to place' child gives no guarantees- all children's development is unpredictable and many adopted babies turn out to have additional needs (as do many unadopted children, but the rates are higher amongst adoptees).
Please don't choose your child by just considering the mechanics I've described. I would adopt in the UK unless you are going to be really hard to match and you are dead set on a baby. The process has flaws, but it gets you ready to be an adoptive parent and it usually matches you with the child you were meant to have. Go into it with an open mind and use the training and assessment to firm up how you feel about age and additional need.
Adopting a kid is an amazing and worthwhile thing to do- I cannot recommend it enough.