Link to the thread in Chat, should it be handy.
Au-Pair:
Typically someone from abroad, EU country these days usually due to immigration rules (can come from a few other countries via Youth Mobility Scheme). Tend to be quite young, though age can vary teen-mid 20's, they are coming to the UK to experience life in the UK, learning English and in exchange for a room and food plus some money (typically under £107 a week), they help around the home doing a variety of household tasks and can include some care of children. Typically they would do around 25 hours a week, so are ideal for parents needing before/after school care. They may or may not have any childcare experience.
Nanny:
They are mostly coming to your home (they can either live-in, or live-out) to provide childcare. They will often also do some additional household tasks like the children's laundry. They will often have a good level of spoken English, though you can recruit from anywhere in the EU, or Youth Mobility Scheme countries, so language level may vary. Nannies may or may not have childcare experience (if it's their first job) but will usually have various levels of experience working with children of various ages. Nannies can be a good option for care of very young children, as you can recruit a nanny who has experience of caring for babies in the past.
DD2 being 7 months old and thus needing full-time care I see as being a problem for you. The spare room may be rather small and if you are recovering from an OP, I'm not sure you would want someone else around the house a lot of the time.
Why is a Childminder not an option? DH leaves at 8pm, so he could leave at 7.30am and drop both children to a childminder, who then takes DD1 to school and cares for DD2 all day. DH then does the pickup at 6/6.30pm ish. If his working hours are fairly well set then I don't see those hours being that hard to find a childminder to accommodate. Have you looked into local childminders - your local council can provide a list.
A live-out nanny may be an option. As a live-out nanny, I start work at 7am and finish at 7pm on my long days. However it's a more costly option than I would expect a childminder would be.
Would someone do it for a few months? That's the question for all childcare provider types... it will vary on the individual person.
I think the first decision you need to make is if you want someone else Living At Your Home.
How much advance notice will you get of the OP date?