I suppose it'll vary a lot country by country. I think most of the UK population will have received at least one dose of the vaccine by the summer and things will seem relatively normal here, and that'll be true of many other countries too. I wouldn't assume it's true of most countries though, even neighbouring countries. The EU countries share the supplies equally but after that point there's a lot of variation in the distribution strategy, the medical infrastructure, population engagement etc.
I'm quite optimistic about the vaccination plan so it's possible you could be offered a vaccine by May, but even with new vaccines possibly coming onto the scene by then, we can safely assume we won't have the supplies for you to get the second vaccination. By that point I wouldn't be surprised to see some form of vaccination passport implemented. You'll have countries at a critical stage in the vaccination program, with light at the end of the tunnel, that won't want to jeopardise that to support tourism. I'd be very surprised if you were able to visit many Asian countries without being vaccinated, and I'd expect most countries to consider you unvaccinated if you've only got one dose.
Even after the entire UK population has been vaccinated (or close to it), I'd still expect there to be travel restrictions on UK travellers in places like Australia (who purchased vaccines later, so will receive them later) or Mexico (who haven't been able to acquire an affordable vaccine yet, and are relying on COVAX vaccinating 20% of the population). We still have very little data on how much the vaccines reduce transmission. If they reduce it a decent amount, somewhere like Mexico might open their borders while a large chunk of the population is unvaccinated, but somewhere like Australia won't. Even if it reduces transmission a lot, but not fully, it's unlikely Australia will allow vaccinated travellers before the vast majority of their population is immunised. This is all the UK are saying about transmission in their
vaccine plan:
I'm sure you'll be able to travel in May, but if you were planning on visiting a lot of countries then I'd expect you to still be wearing masks in most places, there'd be restrictions on crowds in some places, and there'd still be localised flare-ups leading to emergency measures. There will surely be some countries that are still experiencing a national health crisis at that point, but a lot of countries should have things under control by then. Maybe a handful will have vaccinated almost all of the population, but many will have vaccinated the groups most likely to strain the health system and by that point they'll be looking to speed up the economic recovery in every possible way. Risk mitigation measures that carry almost no economic costs (like masks) still dominant but restrictions on activities pulled back a lot.