1.How we are with timing - is this enough time or not to find someone for July?
Should be ok, as some people are available immediately, others may need to give a month's notice. Probably do need to be reasonably quick getting adverts out. Make sure they are realistic in terms of what salary you will offer and advertise it with the Gross pay amount. As the employer you would pay the gross amount but run PAYE, deducting Income Tax and National Insurance plus employee pension from their pay and paying that to HMRC / Pension on their behalf. Nanny payroll providers can assist with that.
2.What are the advantages/disadvantages of going through an agency?
I've not had a job via an agency for a long time now. They should narrow down applicants and only provide you with the best of the best, who meet your requirements as perfectly as possible, but if that happens or not is another matter.
3.If we don't go with an agency, then what's involved in finding a nanny ourselves?
Use a nanny payroll company, they can help with contract of employment, setting up payroll and pension. NannyPaye, Nannytax, Taxing Nannies, PayeForNannies just to name a few.
Advertise on listing sites like Childcare.co.uk - have a photo (note they are small) and make your listing as detailed as possible. Pay for membership so people can contact you and you contact people you search for on the system who look possible candidates. Be very clear about salary and working hours, and that you are working from home as that will put off some applicants.
Local Facebook groups can be good for finding people, if the group allows ads. Ask around anyone you know at groups baby goes to, as other parents/nannies may know of nannies looking for work in the local area.
4.Anything you wish someone would have told you when you were looking for a nanny?
Tell people about car parking... I've been for some interviews recently where there was no thought given to where nanny would park their car and thus how nanny would get a baby and carseat and bags to the car for going out to toddler group, activity group etc.
5.Any other info or tips you think would be useful for us to know
Really know your numbers. Costs can be high and it is not just the salary of the nanny that you need to consider, but also the Employers NI, admin costs, activity costs and travel costs associated with that. Don't start with having a nanny work and then a month or so down the line decide that it is too expensive. If you need candidates who are already Ofsted registered and able to accept Tax Free Childcare scheme, then make that clear in adverts.