You have not mentioned where you want to post the application from but my response below is based on the assumption that you are aware of the fact that Royal Mail is a UK based service (need to be in UK to use it).
I am not sure if Royal Mail Same Day Delivery service is available for retail customers. One needs to have / open an account for this (as per RM website), so I am guessing this service is meant for business customers only. But I might be wrong, so do check. If you find out that you can avail of the Same Day service as a retail customer, you can also use this service to send your application latest by tomorrow.
Alternatively, you can use Royal Mail Special Delivery 1pm service to post the application today. Use the courier address for sending the package. It will delivered tomorrow before 1pm for sure and will definitely be signed for on receipt. Please note that you must use this service today, so the package reaches UKBA by tomorrow (Friday) as receipts at courier address are signed for only on working days (Mon-Fri).
On the other hand, iIf you are going to send the application from outside UK, then use any courier service that can manage to deliver the package by tomorrow (Friday). The receipt will be signed as long as courier is delivered Mon-Fri during office hours.
Just make sure that, irrespective of which service you use to courier your application, you send the application to the courier address and not the PO Box address. Here's
why.
regards