purses/backpacks in museums, etc.

Churches are not that restrictive.
But you should research the rules in advance at specific tourists sites on the day before you visit them or ask concierge before you set out.
It is no different than if you go to a museum or tourist site in Oklahoma for example http://www.okcmoa.com/visiting-faqs/

Purses are generally permitted, but expect them to be inspected at museums.

Backpacks usually have to be checked at museums.

Just to save confusion, "purses" in the UK are like this:
https://www.antler.co.uk/media/catalog/product/cache/1/image_lst_c34efafbb004d8bbea085a25563aae51/1800x/040ec09b1e35df139433887a97daa66f/p/o/poppy-small-coin-purse-with-clasp-pink-internal_3.jpg

I think what you're referring to is a handbag:
http://ecx.images-amazon.com/images/I/71t4DfxMoaL._UL1500_.jpg

Ha ha. Thanks Harleydonski for the clarification. That is the southern part of me coming out. Down here, we call a handbag a "purse." I've already learned something and I haven't even left home. Rick would be proud!

Almost everyone seems to have a backpack these days.
The Sir John Soanes Museum, don't allow backpacks, they have a cloak room, ladies purses have to go in a clear plastic bag to carry with you.

As I said on the similar France thread, some museums are no longer taking large items of luggage because tourists are leaving their luggage there all day, to avoid paying railway station or airport left luggage fees, filling the cloakrooms up to the point that genuine visitors can't check their stuff.

I had a sizable waist pack last year in London. I visited several museums and no one asked me about it. I know I checked it at the V&A b/c I was tired of carrying it.

Considering the Op/CM is from the USA and the forum is as well, then purse would be a correct use of the word. I use the word purse or sometimes, but not usually, the term pocketbook. I've never heard the term handbag used as a term for pocketbook...I would think of carryon luggage. So to the British, adaptation is a must here...although my British son-in-law has never pointed out this language difference...and I've heard many, so maybe it depends on from what part of Britain you hail? ;)

Central NYer here - my grandmother (born in the 20s) called it a pocketbook, but that was a pretty old-fashioned term for that area by the time I was old enough to be aware of such things. My generation (X) called them purses and now that I live in New England I hear purse and handbag pretty equally, with "handbag" seeming to connote a more expensive or fashionable item than "purse."

It doesn’t matter what area of the UK you come from, it's always been called a handbag:

http://dictionary.cambridge.org/dictionary/english/handbag

I carry neither a purse, nor a handbag, nor a pocketbook. But my small backpack was inspected at the British Museum last year. There was a special line for people with backpacks. It was not inspected at the Imperial War Museum.

Did some research as the info was scattered. Here is what I found after emailing the museums below.

BACKPACKS / RUCKSACKS

British Museum
- £1.50 per item, 50 pence for umbrellas, fold-up push chairs free.
- Maximum size: 40x40x50cm (15.7in x 15.7in x 19.6in)
- Maximum weight: 8kg (except fold-up push chairs)
- Main cloakroom is to the west of the main entrance

Victoria & Albert (V&A)
- No charge
- Rucksacks are allowed in the museum however if they are particularly big you may be asked to put them in the Cloakroom
- We also recommend that you do not take them in to exhibitions.

Tate Modern & Britain
- No charge. This service is free.
- Medium/large backpacks must be left in the cloakroom at Tate Britain or Tate Modern.

National Gallery
- Coat: £1*
- Bag (max. 38cm x 30cm / 14.9 in x 11.8 in): £1*
- *Charges do not apply to Members
- Folding bikes, adult scooters, other oversized objects, and group luggage items are not permitted
- We recommend and request that you do not bring luggage and suitcases to the Gallery

National History Museum
- No charge
- We do allow the carrying of backpacks / rucksacks in the Museum
- You can either take them around with you or store them in the cloak room for the duration of your visit

National Portrait Gallery
- The lockers cost £ 1 or the Gallery kindly asks for a free donation for the use of the cloakroom
- Small backpacks are allowed inside the Museum, but not inside the temporary exhibitions
- Large backpacks are not allowed in the whole Museum
- Near the entrance there is a cloakroom and a lockers area where you can leave your backpacks if its size is not allowed in the Gallery

Here is information for vistiting the Houses of Parliament: http://www.parliament.uk/visiting/access/security/

Specifically, regarding bags, the following apply:

Bags and bicycles
Please restrict the size of bags brought to Parliament as there are no facilities for storing or leaving bags or other personal items. Commercial left luggage facilities are available nearby at Charing Cross, Victoria and Waterloo stations.

Folding bicycles, luggage and other items that are larger than 550 mm x 360 mm (approximately the size of airline cabin baggage) will be refused entry unless authorised in advance. Non-folding bicycles may not be brought on to the parliamentary estate. The nearest bicycle rack is on Millbank.