There are two issues at play here, the currency doesnt determine where the account is domiciled. you could have a HSBC Bank plc (the UK bank) USD account (or equally one in EUR) or you could have a HSBC Bank USA NA (the US bank) USD account (or equally a CCF SA EUR account). which could all be opened by HSBC in the UK depending on how and where you trade.
in each case if you were paying a cheque into an account in the UK (whether it be denominated in GBP, EUR or USD) the cheque will be dealt with in the same way, it could be negociated or sent for collection which is the process which can take stupid legths of time.
however if the account into which the item is paid is within the 'domestic' clearing system that it originates from it will be dealt with in the same way as if you were a domestic person/business.
worth noting that collection (at least by HSBC where i work) is often not as painful as it has been in the past, eg rather than posting a cheque drawn on (eg) Wells Fargo to them for them to deal with at some stage, and get round to returning the payment to us, we simply deliver it to HSBC USA who process it through the domestic clearing system, and of course then we can deal with the funds much faster internally once received in the US.
Ian