Ex-EU baggage question

Hi all,
We are flying LHR-BUD and then, the next day, BUD-LHR and LHR-HKG.
Are we able to fly "hand luggage only" to BUD and back to LHR and then pick-up our checked luggage at LHR and complete check-in for the LHR-HKG segment separately, as normal?
Or will we have to check-in at BUD for both the BUD-LHR and LHR-HKG flights and ensure luggage is deposited at BUD and "checked through" at that stage?
Thanks in advance.
We are flying LHR-BUD and then, the next day, BUD-LHR and LHR-HKG.
Are we able to fly "hand luggage only" to BUD and back to LHR and then pick-up our checked luggage at LHR and complete check-in for the LHR-HKG segment separately, as normal?
Or will we have to check-in at BUD for both the BUD-LHR and LHR-HKG flights and ensure luggage is deposited at BUD and "checked through" at that stage?
Thanks in advance.