1. The vaccine doesn't make you immune immediately. I can't remember the exact figures but you get something like 30% protection 2-3 weeks after the first dose and 85+% protection 3-6 weeks after the second vaccine depending on which vaccine you were given. I believe the vaccines are licensed to be given 3+ weeks apart, currently in the UK they are given 8-12 weeks apart so you are potentially looking at 3-4 months from your first jab.
The vaccination program is a medium term solution to the covid pandemic with each wave of the virus there will be a smaller proportion of the community highly susceptible to catching the virus because they don't have the full immunity from the vaccine or they have elected not to have the vaccine.
The UK is heading into a third wave, all those that extremely vulnerable and were vaccinated in the early phases of the vaccination program will now all have the "full" protection from the vaccine (although 5-10 per 100 people in this group exposed to the virus could potentially catch it) so shouldn't contribute significantly to the number of cases. Likewise those with underlying conditions should all now have had both vaccines and should be approaching full protection if they haven't already.
That leaves the unvaccinated, those who so far have only had 1 vaccine and the unfortunate people who are "fully" vaccinated who still catch or transmit the virus to contract and/or become ill with the virus. By the 4th wave there will be a smaller pool of people highly susceptible to catching the virus by which point community transmission should be much easier to control.
I'm not knocking the vaccination program at all. I think it's great but I'm not sure the subtleties about what protection it offers and when have been adequately explained. It certainly plays a role in protecting the extremely vulnerable, the health services and the economy as a whole.
So to your question, should you let him visit if refusing to vaccinate himself. Maybe. Maybe not.
What else is he doing to protect himself, the broader community and your family? Does he generally take the virus and it's consequences seriously
We spent a week with our support bubble (under 1s) a little while ago and a member of the bubble is vulnerable to any infection currently. In the run up to the visit we did a lateral flow test 11 days before we were due to visit giving us time to isolate and get a PCR test should it be positive and then again the day before we left. We now do them at least once a week.
We did a shop that covered 2 weeks of food, 2 weeks before we left so we had no need to go to the shop in the 10 days before our planned departure. We went back a step in the lockdown opening up procedure to add a layer of extra precautions and were even more vigilant about masks, distance between us and other people, avoiding places with a high footfalls and washing and hand sanitising our hands after touching anything outside the home as we didn't want to catch any other bugs not just covid.
We were quite comfortable with this level of precaution and were able to have a lovely visit.
Things are pretty open now and most people in our family circle have been vaccinated at least once, most twice with an adequate delay. If we meet them indoors as currently permitted we are insisting on a negative lateral flow test on the day of the visit (we are also doing them) as well as checking whether they have been particularly reckless.
It helps us all feel more comfortable with visits and allowing parts of our lives return to normal even if some aspects aren't.
We will permit visiting from those that are making sensible precautions with or without vaccinations. Have a think about what would make you feel comfortable to have him visit having not had the vaccine.