How quickly will they obtain a warrant?

It depends on the nature of the crime, among other things.

Under Section 8 of the Police and Criminal Evidence Act 1984 (PACE) the police must provide evidence to a court that a search warrant is necessary to secure and obtain evidence relevant to an ongoing investigation into a crime. If there is evidence that time is of the essence with regards to the evidence at stake (e.g. there is a high probability that the evidence will be destroyed unless it is secured immediately) then the search warrant will be expedited.

Once the search warrant has been secured, it will be an operational matter for the police to decide when and how to act on it. They could choose to do a dawn raid, for example, or they could monitor the suspect for a week and then search the property when the suspect is out.

Be aware that under Section 18 of PACE, a search warrant is not necessarily required. The police may, in the course of arresting someone for an indictable offence, search the premises without a warrant provided certain criteria are met. Of course, this is predicated on the police having grounds for arrest, and in the vast majority of cases, a search warrant will be obtained.