For most vehicles, you should allow roughly 30-60 minutes for replacement depending on how easily the wheels can be removed. If you are replacing all 4 tyres, balancing and pressure checks are done, which can take additional time depending on how busy the shop is.
If you're lucky, you might get your tyres fitted on the same day, but this largely depends on availability of the tyre size locally, as well as the popularity of the tyre size. Taking all this into account, to improve your chances of same-day fitting, be flexible on fitting times, and try to search early in the day for alternative brands to the one you're fitting.
If you experience frequent cold and wet weather, and you want a tyre that performs well in light winter conditions, an all season tyre would be a good option. They maintain a level of flexibility in colder conditions, and give good year round traction in most weather conditions so that most drivers will not have to worry about changing between summer and winter tyre sets.
If you notice a rapid drop in pressure, make a stop and avoid continuing the journey. It can cause permanent damage to the tyre sidewalls. Driving with a flat tyre can cause extensive and irreversible damage to the tyre. Many punctures can be repaired especially in the centre of the tread. However, damage to the tyre shoulder often cannot be repaired. As stated in the vehicle handbook, if you have a spare or a repair kit, do the needful and inspect the tyre at the first possible opportunity.
It's not always necessary, but alignment is highly advised if the steering is off-centre, the car drags, or there is uneven wear on the old tyres. Correct alignment helps new tyres last longer, improves the stability, and can potentially lower the rolling resistance improving the fuel economy.