Graham - there are a few things to consider here and there are a number of threads on the forum that deal with very similar scenarios
At HMLR we can't advise you on what to do in any given scenario until you have decided what is to happen. The options will and can vary but once decided we can assist in explaining what forms are needed and what the process is.
If you start with the actual flat then I assume it is a registered leasehold title; you and your late FIL are joint legal/registered owners; and the 50% share you refer to which relates to your beneficial ownerships, was protected by way of a form A restriction on the register -
Joint property ownership: Overview - GOV.UK
If that's the case then on his death the legal ownership of the property passed to you to deal with and did not form part of his estate. Probate was/is not required in such a scenario where the property is concerned.
His 50% beneficial share still exists and if you have decided that your partner is going to be a new joint legal owner then you would need to transfer the ownership/title to you and your partner using a form TR1 (transfer) and not an AS1 (assent. The AS1 is only used where an executor is transferring to a beneficiary and that is not the case here.
You cannot transfer a half share as the legal ownership is always dealt with as a whole
Our online assisted guidance explains what's needed if that's what you decide to do -
HMLR Guide Page · HM Land Registry
You refer to being 'told that you need to complete an assent' but that's not the case here. You don't state who told you but if it was a legal adviser then do go back and discuss options as appropriate
Land registration is complex, designed to protect legal and financial interests in property. There can be significant consequences for any error. Please consider seeking legal representation. Before making an application without legal representation, It is important to
consider the benefits of using a conveyancer.