I hate to throw a spanner in the works but normally not if you have lost your passport and also the service is not normally possible after 11 on a saturday.
This is what it says on their website.
Travelling within two days
In this case you will need to apply in person at one of our offices using our Premium same day service. There is an extra charge for this service (see How much does it cost?) You need to make an appointment to apply at a Passport Office by calling our Adviceline on 0870 521 0410.
This service is not available if you are applying for a first passport, a full / total name change to your existing passport, extension to a passport, or to replace a lost, stolen or damaged passport.
The Premium service is a guaranteed service whereby you deliver your application and collect your passport on the same day, provided your application is received before the cut-off times of 16:00 in London, 13:30 in Liverpool, 13:00 in Peterborough, 12:30 in Glasgow and Newport and midday in Belfast and Durham. On Saturdays the service is available to applicants with appointments before 11am.
If you want someone to collect your passport for you, they must provide evidence of their identity and a signed letter from you giving them permission to collect your passport. If the application is for a child, the person who has signed the section 9 of the form must sign the letter. We can only issue a passport if the person applying is in the UK at the time they apply.
Conditions of this service
The guarantee for this service applies only if your application is straightforward and if you have completed the form properly and supplied all the necessary supporting documents. If you supply incomplete information or we cannot confirm your eligibility from the documents you have given us, we may not be able to issue a passport in the expected time.
We strongly advise that you do not book any travel arrangements until you have received your new passport.
Please note:
currently demand for appointments throughout all IPS Regions is exceptionally high, and you should therefore allow at least two weeks for your appointment date.
if you make an appointment please arrive no more than 10 minutes before your appointment time. You will not be seen any earlier and your admission will be delayed until the appropriate time. This is to ensure our counters are as pleasant as possible for all our visitors.
customers should be aware that all callers to our counters might be screened on entry, to ensure the safety of our staff and other customers. Any item considered to be of a potentially dangerous nature might be removed for safe keeping before entry to our premises is allowed.
Having said all that if you don't ask you don't get so you may as well try. Good Luck