I am sorry to hear that your Mum has died - it is a hard time for you.
Two things are going on here. Firstly you are worried about your Dad, and secondly you have your own grief to handle.
I think that your hope that all of this will not impact on your daily life is a vain one. This is a lot to deal with and you need to let it run its course and accept that life will be very different for a while. That this is normal.
I can see this from the other side of the fence. My OH died in 2020 and it was of course devastating - I had nursed him for a long time and watched him change and deteriorate in every possible way. It was a very tough time. I was not in a good place for quite a long time. October is only a short few months past and I am afraid that your Dad will be processing his grief for some while to come.
The fact that he is going out and continuing to socialise is a big plus; but please remember that doing all that will be a massive effort for him. He will be holding back the tide when he is in company and it is inevitable that it will spill over when he is talking to his family and those closest to him.
It might help you if you accepted that his grief and his responses are entirely normal. That might help you to be at peace with them. You say: He’s crying every day, every phone call and I’m overwhelmed and stressed with his grief. You cannot change his crying or make it go away; but you can change your response to it. Your response is to feel overwhelmed and stressed by it. But it does not have to be that way - you can choose to approach it in a different way - to see it as normal and inevitable at this stage and to embrace it rather than wanting it to simply go away. I used to tick along with a brave face and everyone thought how marvellously I was coping - but my DDs knew at what cost. I would ring them beside myself with sobbing and grief, and one of them would pop over and simply let me talk - about my OH, about the awful last months, about my guilt that I might not have been the best carer etc. Sometimes we would share happy memories, and sometimes we would cry together and share our grief.
You do not have to be the stalwart prop for your Dad all the time - you are allowed to cry with him when he does, so he can support you too. 60 years is such a long time and his every body system and thought will be adapted and habituated to having your Mum around - that huge void is so hard to get used to. And to be honest older people find it harder to adapt to change of any sort, let alone something as massive as this.
Are you getting good support from your partner? I think this is pretty important. I knew that when my DDs had been mopping me up, they had strong loving partners (who were also missing OH) who were there to mop them up too. Your relationship will be impacted; and your work; and your own well-being...... because this is what grief does. Your work colleagues will also have suffered losses - it is simply part of the human condition - and they should not expect you to be all bright eyed and bushy tailed.
I think you are hoping for too much in wishing that this major life event will not have an impact on you and how you relate to others for a while to come.
I found the best way forward for me was to be completely open: in company I talked about him dying (never passing) and about his life and about the challenges I was (and still am) facing - now and again I shed a tear and friends just gave me a hug.
You are being a splendid daughter with all the support you are giving him - but do not hope for too much too soon. 