the shortest answer is yes and yes .... BUT
//I was wondering if it's possible for hospitals to know if you have an arrest warrant?//
yeah if someone tells them - like you go in chained to a policeman. No in principle - that is there is no bat-line red phone which rings and says....
Hospital information is confidential
and doctors are chucked off the register of doctors if they go against the strict rules ( as in happening as we speak - recording a pt without consent was taken as breach of confidentiality even tho there are CCTV cameras in non pt areas in hospital. ( GMC v Dr Natalie Blakely ) )
BUT
there are various duties -
statutory duties - the hospital has to inform the police if a patient comes in with - a stab wound, explosives and terrorism I think
and under court rules
1) faced with a court order
2) subpoena'd and told to bring records
//Do the police ask hospital staff to call them if a certain person of interest comes in? //
yes they may - this happened where I worked and the staff refused ( 1990) so they staked out the hospital and waited without a warrant. and other parents complained there were a lot of plain clothes police in a childrens hospital.
we were told we would be interfering with an arrest - very bad for ones career - but no one took the point that at the time they were waiting, the future arrestee was NOT in the hospital, so it was not possible to interfere with an ....
Records are used only for names and addresses and never for details of smelly dongers or whatnot
and finally
http://www.bbc.co.uk/news/uk-politics-44054516
where immigration officials had access to addresses
line 2 : NHS workers very unwilling as it deters sick people from attending hospital
well you did ask