Typically, the following timescales apply:-
.uk Domains - once the IPS TAG change request has been processed by your existing registrar, your domain can be transferred into your account immediately.
gTLD (.com, .net, etc) Domains - There's a slightly longer process to transfer these domains over, the following steps are taken:
1: An authorisation email will be sent within a few hours of submitting the transfer request from 'no-reply@mainnameserver.com' to the admin contact on the domain as per a whois report. (you can check a whois report here: http://whois-search.com/)
2. The new registrar asks the owner to provide the authorisation code.
3. The old registrar verifies that the authorisation code provided by the new registrar is correct. The transfer will usually fail if no correct authorisation code is supplied.
4. The old registrar asks the owner to confirm that they want to transfer. If the domain owner does not respond within 5 days, the transfer is completed at that point. If the owner does respond affirming the transfer, it's completed more or less immediately.
For registries which do not support authorisation codes, the new registrar may ask for simple approval from the owner, and the old registrar will ask for approval from the owner, without which the transfer will fail after a number of days
Typically, a transfer of a gTLD domain should complete within 5-7 days.
Other Domains - Transfers typically take up to 7 days to be processed.
Please note that if you are switching nameservers, DNS propagation times will apply - this is usually 3-6 hours, but can potentially be up to 48 for all users.