Career nanny viewpoint:
Some childminders can offer overnight care so I wouldn't completely discount this option, although they absolutely must have express permission from Ofsted to do this (for children under eight). In saying that, your young son will certainly benefit from the familiarity of his own home and bed, especially if your absence might cause some anxiety.
Some nanny agencies can charge a hefty fee, you might do better to look at platforms such as childcare.co.uk and consider recruiting yourself (or at least establish a pool of qualified leads that you can pass to a recruiter to screen for a fee). If you go this route, PM me and I will give you a checklist of what to look for. However, whist this can turn up the odd miracle, many nannies are happily employed and may lack the time capacity to assist you. In saying that, you might be lucky with someone who can fit you around their current role with early-morning/late night cover in months to come, but nanny-shares are a minefield during current Covid-19 times!
Friends and connections - always an excellent way to start. Put the words out on your networks, speak to people who already have nannies, and spread the work about your potential opportunity. Whilst it is ultimately your prerogative to check the credentials of applicants, a few words of endorsement (or polite caution) can me most useful.
I would absolutely look to take someone on a an employed basis, HMRC have been known to identify domestic staff that are locally 'deemed' as employees, then pursued the 'employer' to recover unpaid NI and Pension contributions. I personally work under both hats, but have a clear audit trail to protect all parties.
The costs vary by geographic location, the qualifications and experience of the provider, and may be slightly weighted for add-hoc hours. In the home counties, £13-15 gross per hour for a higher-end nanny, and £10-12 gross per hour for less qualified/experienced. Lesser costs in Essex, the South-West and Northumberland.
Be very careful not to confuse Nett and Gross wage proposals!
Finally, you would need to consider the minefield of finding someone that 'fits' with your family set-up. This is not usually an issue for experienced professionals who will step-in and immediately go to task, but can be a challenge with cheaper and less experienced operatives. Consider involving your son in this recruitment process, ask him to list the kind of things that would be important to him, and those that will make him feel safe and happy. You can then pick-out his matching points to help him with the transition once you settle on a solution.